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For all integers r, s>2, there are infinitely many primes which can be written as the sum of an r-gonal number and an s-gonal number. In general, any linear combination of an r-gonal number and an s-gonal number with coprime positive integer coefficients produces infinitely many primes. However, unless r=s=4, such primes can not be described by finitely many congruences.
Bhattacharya et al. (Sat,) studied this question.
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