In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $τ$ avoiding $\mathbb{F}$-stopping times and such that $\mathbb{F}$ is immersed in $\mathbb{G}$. As an application of this, we can solve an exponential utility maximization problem in the enlarged filtration $\mathbb{G}$ following the dynamical approach, based on suitable BSDEs, both over the fixed time horizon $[0,T]$, $T>0$, and over $[0,T\wedgeτ]$.