You might be interested in extensions to the Sylvester Schur theorem, which by your constraints shows that c is bigger than k^2 as the set of consecutive integers in the product must have a single multiple of q^2 for some prime q bigger than k. A paper of Saradha and Shorey from 2003, Almost Squares and Factorizations in Consecutive Integers, shows the sparsity of solutions to your equation where k-1 of the numbers on the right hand side multiply to a square.  This may be useful for you in a citation search .

Gerhard "Not Quite Almost Powerful Numbers" Paseman, 2020.07.25.