Does potential at a point in circuit allows the flow of charge? Suppose I have the following circuit with lamp connected and there's also a really long neutral ( total charge zero ) conducting wire under the circuit as follows :

Then , from the theory of electromagnetism, it is known that the potential at distance  $ r $ from the conducting neutral wire ( colored in black ) is: $ V(r) \propto \lambda ln(r_0/r) $ where $r_0$ is some reference radius and $ \lambda $ is the charge density of the wire.
So, since there exists a potential from the wire, then the circuit will 'feel it' and charge will flow in the circuit for a brief period of time as shown in the picture below:

The charge flows this way because the potential is higher when we get closer to the conducting neutral wire ( colored in black in the previous image ) and is lower when we get further away from the wire ( so charge flows from higher potential to lower potential ), in addition, charge carriers in the circuit loop rearrange themselves, in such a way that the field inside the conducting loop is zero. This happens for extremely short time. 
So there exists a charge flow in the circuit, but such charge flow won't turn the lamp on because the total current in the circuit will be zero ( since the superposition of two currents with equal magnitude but different directions will give zero ) ...
Ok then, in this specific example the total current in the circuit is zero...
But what if there was a different neutral object ( not neccesarily a wire ) creating a potential field such that the total current in the circuit is not zero? could such situation even exist? if it does, then how come electric stuff dont just get turned on randomly at real life ( we wouldn't need batteries and etc )? is it perhaps because the current\potential produced is too low?
 A: No, this isn't possible. This can be seen from Maxwell's Equations, specifically the law of induction (in integral form):
$\oint_{\delta\Sigma}\mathbf{E}\cdot{d\mathbf{l}}=-\frac{d}{dt}\iint_\Sigma \mathbf{B}\cdot d\mathbf{s}$
This effectively states that the only way that an electromagnetic field can generate a moving current in a closed loop of conductive material is if the flux of the magnetic field through the loop changes with time. Therefore, it is not possible for a static electric/magnetic field to generate a moving current in a looped conductor.
Edit: A question was asked in the comments about the case where you have a battery in the circuit - why that creates a current, if the situation is static?
The case with a battery is dealt with by a different part of Maxwell's equations. The law of induction (above) applies only to currents that are generated by an external EM field. Batteries are dealt with by Gauss' Law:
$\nabla\cdot\mathbf{E}=4\pi\rho$
This states that an electric field can also be created by objects that carry a charge (i.e. charge carriers/electrons). The battery generates an electric field (potential difference) around the circuit, because it has an imbalance of charge carriers between the positive/negative terminals. This is different to the case of a current being induced by an external EM field.
A: 
But what if there was a different neutral object ( not neccesarily a wire ) creating a potential field such that the total current in the circuit is not zero? could such situation even exist?

Yes.  Electromagnetic induction can create problems for some circuits (like introducing interference into a radio receiver or amplifier).  

if it does, then how come electric stuff dont just get turned on randomly at real life ( we wouldn't need batteries and etc )? is it perhaps because the current\potential produced is too low?

Unless you get really close, or you get favorable geometries (like coils), the induced current is truly tiny.  
