The distinction between primary and secondary constraints is not a very fundamental one.
Once we have gone over to the Hamiltonian formalism, we can really forget about the distinction between primary and secondary constraints.
[1] In Hamiltonian mechanics, a primary constraint is a relation between the coordinates and momenta that holds without using the equations of motion.
A few authors use more refined terminology, where the non-primary constraints are divided into secondary, tertiary, quaternary, etc.
Primary and secondary constraints were introduced by Anderson and Bergmann[4] and developed by Dirac.