Is it true that for all $n>N$ that n is the sum of two or more distinct primes that are either large or (for parity reasons) 2?

I feel like I've seen a result allowing this with $p\gg n^e$ for reasonably small e (0.4?). In my case I could be much more lax: even $p>23$ would suffice. But I'd like an effective N, actually very small if possible ($N=10^7$ would be ideal).

I stress that the number of summands need not be small; I'm not trying anything like Goldbach. Seventeen primes or even two hundred would be fine.