I've recently been looking at the quantum mechanical description of a charged particle in an EM field and have come across this classical Hamiltonian: $$H = \cfrac{(\mathbf p - q\mathbf A)^2}{2m} + q\phi \, .$$
I can interpret the second term as the electric potential energy of the charged particle due to the $E$-field. Is there a similar, physically intuitive, way to understand why there is a correction of $-q\mathbf A$ to the kinematic momentum $\mathbf p$?

I have managed to obtain this Hamiltonian from the Lagrangian for the system, so it's just the intuition that I'm struggling with.