Generally speaking, if they ask us to prove that there are infinitely many numbers that can't be written in a certain way, how should we try to solve the problem? I've never seen a solution to such a problem yet, so I can't think of any strategy to attack the problem in the title. I think the problem leaves too many possibilities too check! If they tell me that an integer number can't be written in the form $a^2+p$ where $p$ is prime that doesn't give me any clues about how such numbers could be. I mean it doesn't put enough restrictions on the numbers that aren't of that form that I go after studying the properties of such numbers.
There is also another problem of the similar type in my book: Prove that there are infinitely many odd numbers such that they can't be written as the sum of less than 3 prime numbers.