This question is very similar to Is it decidable to check if an element has finite order or not? and has the same answer. Namely, an example was constructed by McCool. He also constructs in that paper a finitely presented group with solvable word problem where you cannot decide if an element has finite order, known as the order problem. I'm pretty sure that the power problem, which is what you asked for here has also been answered before on Mathoverflow.

Note the group is explicitly constructed via a recursive presentation and then uses Higman embedding to get the finite presentation so it may not be as explicit as you would like because you would have to chase the Higman embedding but the word $w$ is explicitly given and fixed modulo the Higman embedding.