**Question edited in view of the comments below**

By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.

My question is,

Question 1.Let $p_1,\ldots,p_k$ be the first consecutive odd primes and $n>\max(e^{e^{36}},p_k)$ be an even integer such that for all $i\in \{1,\ldots,k\}$, $n-p_i$ is composite. Is it possible that $n=p_r+p_sp_t$ (where $p_r,p_s,p_t\in\{p_1,\ldots,p_k\}$)?

Note that it is easy to show that if $n>p_k+p_k^2$ then the answer to the above question is "yes". So we only need to check whether the answer to the above question is "yes" for all even integers $n\le p_k+p_k^2$.

Following Gerhard Paseman's suggestion below I think a possible approach of attacking the problem will be to know more about the integers $n$ for which the hypothesis of the question holds. In this regard I have the following specific question in mind,

Question 2.Let $p_1,\ldots,p_k$ be the first consecutive odd primes and $n$ be an even integer such that for all $i\in \{1,\ldots,k\}$, $n-p_i$ is composite. What is known about the lower bound of $n$ (in terms of $p_1,\ldots,p_k$ satisfying such hypothesis?

Based upon my calculations, my best guess is that that $n> p_{k+1}$ but I couldn't prove it.