Physical meaning of the difference between kinetic and generalized momentum in electromagnetic fields

I recently came across the fact that, for a charged particle in an electromagnetic field, the generalized momentum differs from the kinetic momentum. Specifically, the generalized momentum is given by $p=mv+qA,$ whereas the kinetic momentum is just $p_{kin}=mv.$ I understand the mathematical origin of this distinction from the Lagrangian and Hamiltonian formulations of mechanics. However, I'm wondering if there is a deeper physical interpretation of this difference-perhaps in terms of energy, work, or some other fundamental concept. Why does the electromagnetic potential $A$ appear in the momentum, and what does that mean physically?