The short answer is "no", the Prime Minister is not required to give an answer that satisfies the person asking the question.
There is an expectation (not an obligation) that the PM attend the commons every Wednesday. It is now a standard part of the Parliamentary week, and most PMs relish the opportunity it gives to perform and communicate with both the Commons and the wider public.
Similarly there is an expectation that the PM will address the question asked. However, there is no requirement for the question to be "answered".
It is very common for opposition MP to ask loaded questions: "Considering X,Y,Z does the Prime Minister agree that the government is a shambles?" Well of course the PM doesn't agree, but that is not the point of the question. Also MPs might ask questions that the Prime Minister doesn't (and cannot) know the answer to. Or questions that he does know the answer to, but won't give a straight answer to, or for which there is no reasonable answer "Will the PM stop this damaging Brexit?" (is the answer "No, I will continue the damaging brexit?")
However if the PM doesn't answer there are 600+ MPs who can point this out, and there is the court of public opinion. So there is no need for the Speaker, or anyone else, to judge if a question has been answered. Questions to the Prime Minister is public and we can all make our own judgement on the PM's performance each week.