A ternary additive problem

The problem of representing a large integer $n$ in the form $n=m^2+x^3+y^5$ has been studied by a number of authors in the past decades. In this paper, we restrict $m$ to square-free integers, and $x, y$ to primes, and show that there is such a representation for all $n\le N$ with at most $O(N^{1-\frac{1}{45}+\varepsilon })$ exceptions. We also improve the recent results of Liu (Acta Math Hungar 130(1–2):118–139, 2011) and Bauer (J Math 38(4):1073–1090, 2008) on related problems.