I wrote a little template that enabled participants to derive an average based on the answers to four multiple-choice questions (each with a numerical percentage attached that could add up to 100%). My question is that is it mathematically correct to then average the average scores?

Thanks

You have an average score for each participant, and then are averaging over all participants.

Sure that would give you the average over all participants and I see no problem with doing that, as long as you aren't averaging the same thing twice.