This is a typical electromagnet.
Let's refer to your figure and say that $g$ is the width of the gap and $A_c$ the section of the core.
If $g << \sqrt A_c$,the magnetic field $\vec B$ inside the core will be approximately the same as the magnetic field $\vec B_0$ in the air gap (this is because in this case the magnetic field lines will stay approximately parallel to each other in the air gap and the flux $\Phi = B \ A_c$ will be conserved).
If this is the case, you will find using Ampere's law that
$$B = \frac{\mu_0}{g} (N i - H l_c)$$
with $l_c$ mean core length.
If moreover $\vec H = \vec B / \mu$, then
$$B \left( 1 + \frac{l_c}{g \mu_r} \right)=\frac{\mu_0 N i}{g}$$
Notice again that this value is the same in the air gap and inside the core (solenoid included)!
For most ferromagnetic materials, $\mu_r$ is of the order of $10^3-10^5$ (https://en.wikipedia.org/wiki/Permeability_(electromagnetism)), so we can approximate the previous relation neglecting the term $\frac{l_c}{g \mu_r}$:
$$ B \simeq \frac{\mu_0 N i}{g} > \frac{\mu_0 N i}{l_c}$$
since of course $g < l_c$. So the field is indeed stronger than the field inside an empty solenoid of length $l_c$ with the same number of turns $N$.
