Yes, I believe the assumption is saying that a graph has at most 5 vertices, one of which has degree 2.Are they saying 5 or fewer vertices AND 1 vertex with a degree of 2?

Let v be the vertex of degree 2, and let v1, v2 be the vertices it is connected to. The hint makes sense to me provided v1 and v2 are not connected: then v can be eliminated and an edge (v1, v2) can be added to create a homeomorphic graph. I am not sure how to proceed with the proof if v1, v2 are connected.