  Simple upper and lower bounds are established for the integral
$\int_0^x\mathrm{e}^{-\beta u}u^\nu t_{\mu,\nu}(u)\,\mathrm{d}u$, where $x>0$,
$0<\beta<1$, $\mu+\nu>-2$, $\mu-\nu\geq-3$, and $t_{\mu,\nu}(x)$ is the
modified Lommel function of the first kind. Our bounds complement and improve
on existing bounds for this integral, by either being sharper or increasing the
range of validity. Our bounds also generalise recent bounds for an integral
involving the modified Struve function of the first kind, and in some cases
more a direct approach lead to sharper bounds when our general bounds are
specialised to the modified Struve case.
