A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $χ_{\rm pcf}^h(G)$ denote the minimum $k$ such that such a colouring exists. We show that for every fixed $h\ge 1$, every graph $G$ of maximum degree $Δ$ satisfies $χ_{\rm pcf}^h(G) \le hΔ+ \mathcal{O}(\log Δ)$. This expands on the work of Cho et al., and improves a recent result of Liu and Reed in the case $h=1$. We conjecture that for every $h\ge 1$ and every graph $G$ of maximum degree $Δ$ sufficiently large, the bound $χ_{\rm pcf}^h(G) \le hΔ+ 1$ should hold, which would be tight. When the minimum degree $δ$ of $G$ is sufficiently large, namely $δ\ge \max\{100h, 2000\log Δ\}$, we show that this upper bound can be further reduced to $χ_{\rm{pcf}}^h(G) \le Δ+ \mathcal{O}(\sqrt{hΔ})$. This improves a recent bound from Kamyczura and Przybyło when $δ\le \sqrt{hΔ}$.