On the Diophantine equation $X^2+p^m=p_1^{n_1}{p}_2^{n_2}$

We consider the Diophantine equation $$X^2+p^m=p_1^{n_1}{p}_2^{n_2},$$ where $p,p_1,p_2$ are distinct odd primes. Using factorizations in quadratic fields, sharp bounds for linear forms in two $p$-adic logarithms, elementary methods, and known results on exponential Diophantine equations, we obtain explicit bounds for $m,n_1,n_2$, and consequently the finiteness of the number of solutions for fixed $p,p_1,p_2$.