Less than $2^{\mathrm{/}}omega$ many translates of a compact nullset may cover the real line

We answer a question of Darji and Keleti by proving in $ZFC$ that there exists a compact nullset $C_0\subset \text{RR}$ such that for every perfect set $P\subset \text{RR}$ there exists $x\in \text{RR}$ such that $(C_0+x)\cap P$ is uncountable. Using this $C_0$ we answer a question of Gruenhage by showing that it is consistent with $ZFC$ that less than $2^{\omega }$ many translates of a compact nullset cover $\text{RR}$.