Based on John Omielan's outline, I will prove below that every integer $a\geq 122$ has a representation with exponents $s_k\in\{1,2\}$.

Let us use the [fact][1] that every integer exceeding $33$ is a sum of distinct triangular numbers. Given this fact, we consider $a\geq 68$, and we define $c$ to be the largest nonnegative integer such that
$$f(c):=\sum_{k=0}^c(k+2)$$
satisfies
$$f(c)\equiv a\pmod{2}\qquad\text{and}\qquad f(c)\leq a-68.$$
Then
$$f(c+2)=f(c)+(2c+7)>a-68,$$
that is,
$$68\leq a-f(c)<2c+75.$$
By the quoted fact, there are distinct nonnegative integers $k_1,\dotsc,k_m$ such that
$$a-f(c)=\sum_{\ell=1}^m (k_\ell+1)(k_\ell+2).$$
We can ensure that no $k_\ell$ exceeds $c$ by imposing the condition
$$(c+2)(c+3)\geq 2c+75,$$
which is certainly satisfied for integers $c>6$. Since we have already agreed on $f(c+2)>a-68$, it suffices to assume that $a-68\geq f(8)=54$, that is, $a\geq 122$.

To sum up, every integer $a\geq 122$ can be written as
$$a=f(c)+\sum_{\ell=1}^m (k_\ell+1)(k_\ell+2),$$
where the numbers $k_\ell\in\{0,\dotsc,c\}$ are distinct. Hence if we define
$$s_k:=\begin{cases}
2,&k\in\{k_1,\dotsc,k_m\},\\
1,&k\notin\{k_1,\dotsc,k_m\},
\end{cases}$$
then we obtain
$$a=\sum_{k=0}^c(k+2)^{s_k}.$$

  [1]: https://proofwiki.org/wiki/Integers_not_Sum_of_Distinct_Triangular_Numbers