Positive part function of random variable

Question

$\{\boldsymbol{X}_i\}_{i = 1}^N$ is i.i.d. and nonnegative random variables with finite mean $\mu$ and variance $\sigma$, and $a>0$ is a constant. How to bound the gap: \begin{equation*} \mathbb E\left[\sum_{i = 1}^N(a-X_i)^+ \right] - \mathbb E\left[\sum_{i = 1}^N(a-X_i)\right]^+ \end{equation*}