Let c be a square-free positive integer and p a prime satisfying p∤ c. Let h(- c) denote the class number of the imaginary quadratic field Q(-c). In this paper, we consider the Diophantine equation cx2+p2m=4yn,x,y≥1,m≥0,n≥3,gcd(x,y)=1,gcd(n,2h(-c))=1,and we describe all its integer solutions. Our main tool here is the prominent result of Bilu, Hanrot and Voutier on existence of primitive divisors in Lehmer sequences.