Charge produces which one: Field or Potential Is it correct to say that when a charge is at rest, it produces field and potential, or does it produce potential and because of the potential dfference, there is a field? Are both of these mathematical constructs for our understanding, or is there any other explanation for electric effects seen around charges?
 A: A stationary charge produces a electric field in space. We know this, because other charges experience a force as a consequence of this field.
This field is special in that, the force due to this field does no work in any closed path. This field is conservative. As a consequence of it's conservative structure, it is possible to associate with each point a (unique) scalar potential value. We can then describe the vector field, in terms of the scalar potential.
A: Summary
The answer is none. A charge placed in space exerts a force on all other charges. This was the most fundamental and the primary observation made when electrostatics was discovered. The mathematical relation is given by
$$\mathbf F_{12}=\frac 1 {4\pi\varepsilon_0}\frac{q_1q_2}{|\mathbf r_{12}|^2}\:\hat{\mathbf r}_{12}$$
where $\mathbf F_{12}$ is the force on charge 1 due to charge 2, $\mathbf r_{12}$ is the position vector of charge 1 with respect to charge 2, $\varepsilon_0$ is an experimental constant and $q_1$ and $q_2$ are the charges of charge 1 and 2 respectively.
Electric field lines
To visualize this better, we used the imaginary concept of field lines, which was then generalized to other vector fields as well. The reason why I am calling field lines imaginary, is because, in reality, there are no field lines filling our space. There are just charges who either attract or repel each other. Field lines are just an imagination tool which immensely helps us in visually analyzing a scenario in electrostatics.
Electric potential
Electric potential is, first of all, arbitrary, since you always need a reference point (where the electric potential is $0$) to define the electric potential at other points. Thus there is no absolute value for electric potential. But assuming that the potential is zero at infinity, let's continue our analysis. The electric potential at a point A is often expressed as the work done by the external agent to bring the charge from $\infty$ to that point A. This might make you feel that, after all, electric potential is something physical. Well, it isn't. Why? Because the work done, itself isn't any physical thing, it's just a definition ($\mathrm d W=\mathbf F\cdot\mathrm d\mathbf S$). You cannot really physically attribute the work done to some other real thing. The same goes for energy. To quote Feynman (from "The Feynman's Lectures on Physics, the New Millennium edition"):

...The law is called the conservation of energy. It states that there is
a certain quantity, which we call energy, that does not change in the manifold
changes which nature undergoes. That is a most abstract idea, because it is a
mathematical principle; it says that there is a numerical quantity which does
not change when something happens. It is not a description of a mechanism, or
anything concrete; it is just a strange fact that we can calculate some number and
when we finish watching nature go through her tricks and calculate the number
again, it is the same.

(emphasis mine)
The above lines were said in the context of energy conservation, however, they apply here as well.
Conclusion
Note: The following discussion may sound a bit philosophical/metaphysical, but I see no reason for it to be incorrect.
So, both, electric fields and electric potential, are man made concepts. They aren't physically ingrained in our reality. Fields and potentials are good way to model our reality, but they aren't the "reality". If you throw away (conservative) forces, you could equivalently describe the whole world with fields and potentials, no problem. But are these fields and potentials anything more than a mathematical construct? No.
To be honest, even forces can't really be said to be ingrained in our reality. It is primarily acceleration which is more physical and measurable, and using this acceleration, we define forces. Here, I am using the word physical for the things which manifest themselves in reality. I don't think we really see fields and potentials manifesting themselves. What we see is a body accelerating, from which we conclude that there must be a force acting on it, from which we, again, conclude that there must be field lines in that region of space (given the force is a conservative on), and so there needs to be a potential associated to it as well.
You see, it's not really the fields and potentials which are felt by us directly, it's always the fundamental properties (velocity, acceleration, spin, etc.) which we measure.
A: 
Is it correct to say that when a charge is at rest, it produces (a) field

From the standpoint of classical electrodynamics, yes. A charge at rest produces an electric field. The electric field produced by a point charge $q_1$ at a location 2 where $r$ is the distance between $q_1$ and location 2 and $\hat{\mathbf a}_{12}$ is a unit vector directed from 1 to 2 is
$$\mathbf E=\frac 1 {4\pi\varepsilon_0}\frac{q_{1}}{r^2}\:\hat{\mathbf a}_{12}$$
The force experienced by a test charge $q_2$ placed at location 2 due to the field produced by $q_1$ is then by Coulomb's Law:
$$\mathbf F=q_2\mathbf E=\frac 1 {4\pi\varepsilon_0}\frac{q_{1}q_2}{r^2}\:\hat{\mathbf a}_{12}$$

Is it correct to say that when a charge is at rest, it produces
(a) potential,

An electric potential is the amount of work needed per unit charge to move the charge from a reference point to a specific point inside the field without producing an acceleration. As already pointed out, the reference point can be arbitrarily chosen.
If there were no charge $q_1$ at point 1 in the above example (nor any other charge producing an electric field at 2), no work would be required to move a charge $q_2$ from 2 to 1. The potential difference would be zero. But with a positive charge $q_1$ fixed at location 1, work would be required to move a positive charge $q_2$ from 2 to 1.

or does it produce potential and because of the potential dfference,
there is a field?

Reverse it.
In order to have a potential difference between two points, an electric field is required to create a force on a charge between the points. Electrostatic fields are produced by electric charge.

Are both of these mathematical constructs for our understanding, or is
there any other explanation for electric effects seen around charges?

To quote from Professor Richard Fitzpatrick, Professor of physics at the university of Texas:
"Incidentally, electric fields have a real physical existence, and are not just theoretical constructs invented by physicists to get around the problem of the transmission of electrostatic forces through vacuums."(http://farside.ph.utexas.edu/teaching/302l/lectures/node17.html#:~:text=Incidentally%2C%20electric%20fields%20have%20a,of%20electrostatic%20forces%20through%20vacuums.)
The so called electric field lines drawn around charges are not, however, physical entities as pointed out by @FakeMod. Some mistakenly think, for example, that the space between electric field lines of a diagram means there is no field between the lines. The density of the field lines simply allows you to compare the relative strength of the electric fields of different areas of the same drawing, the greater the density the greater the relative strength of the field and therefore the greater the force experienced by a charge placed in the field.
A better (than field lines) way to show the strength of the field would be using various shades of gray over an entire area, the darker the shading the greater the strength. However, shading would not convey information on the direction of the electric field at a given location.
The arrows on the field lines, however, conveys information on the direction of the field along the line. By convention, the direction of the field is the direction of the force that a positive charge would experience along the line.
The use of field lines of various density and arrows is a compromise.
Hope this helps.
A: Everything we talk about, like charge, acceleration, force, field, potential, etc. is just part of the mathematical framework that we choose to describe observations. You could, for instance, start with quantum fields for electrons and photons and never discuss the idealized motion of macroscopic compound objects.
But once you choose a theoretical framework like classical electrodynamics, every derivable concept is there with the same right, and it is a matter of presentation which one you define first, and which one is derived later. You do your math, name things, think of measurements, but in the end, all theories have their limits of applicability where you will say: Actually you have to see it all differently, and start from scratch with other concepts that only under special conditions approximate the old concepts.
Fun fact: In quantum electrodynamics (QED), physical states with a fixed number of energy quanta (photons) do not even have a well defined electric or magnetic field strengh. And electric and magnetic field (E and B) are complementary like position and momentum in Heisenberg's uncertainty principle. Everything is only as real as it is useful to explain a given situation.
