A refinement of a result of Andrews and Newman on the sum of minimal excludants

In this article, we refine a result of Andrews and Newman, that is, the sum minimal excludants over all partitions number n equals into distinct parts with two colors. As consequence, find congruences modulo 4 8 for functions appearing in refinement. We also conjecture three further these functions. addition, initiate study $$k\mathrm{{th}}$$ moments excludants. At end, provide an alternate proof beautiful identity due to Hopkins, Sellers, Stanton.