A065377 (in the online encyclopedia of Integer Sequnces) the list of primes which can't be written in the form $P+K^2$ (p prime and k>0 being an integer), and it's 2,5,13,31,37,61,127,379,439,571,829,991,1549,3319,7549. The comments say that this list is "probably" finite with 7549 being the last entry.
A possible naswer to the question if this list is finte maybe in this paper ---> ARTICLE
This conjecture states that every large integer n that is not a perfect square is the sum of a prime and a square, and predicted an asymptotic formula for the number of such representations. If true, this immediately implies the list is finite.
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