  Let $\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as
the sum of a prime and a $k$-th power. Define E_k(X) := |\{n \le X, n \in I_k,
n\text{not a sum of a prime and a $k$-th power}\}|.
  Hardy and Littlewood conjectured that for $k = 2$ and $k=3$, E_k(X) \ll_{k}
1. In this note we present an alternative approach grounded in the theory of
Diophantine equations towards a proof of the conjecture for all $k \ge 2$.
