Let $0 < β\leq α< 1$ and $κ> 0$. I prove that there exists $η> 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = α$ and $\dim_{\mathrm{H}} B = β$: $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq α+ η\} \leq \tfrac{α- β}{1 - β} + κ.$$ This extends a result of Bourgain from 2010, which contained the case $α= β$. The paper also contains a $δ$-discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form $a_{1}B + \ldots + a_{n}B$.