Secondary terms in the first moment of $\left|\operatorname{Sel}_2(E)\right|$

We prove the existence of secondary terms of order $X^{3 / 4}$, with power saving error terms, in the counting functions of $\left|\mathrm{Sel}_2(E)\right|$, the 2-Selmer group of $E$, for elliptic curves $E$ having height bounded by $X$. This is the first improvement on the error term of $o\left(X^{5 / 6}\right)$, proved in [11], where the primary term of order $X^{5 / 6}$ for this counting function was obtained.