On functions whose graph is a Hamel basis

We say that a function $h \colon \mathbb{R}\to \mathbb{R}$ is a Hamel function ( $h \in {\rm HF}$) if $h$, considered as a subset of $\mathbb{R} ^2$, is a Hamel basis for $\mathbb{R} ^2$. We prove that every function from $\mathbb{R}$ into $\mathbb{R}$ can be represented as a pointwise sum of two Hamel functions. The latter is equivalent to the statement: for all $f_1,f_2 \in \mathbb{R} ^{\mathbb{R} }$ there is a $g\in\mathbb{R} ^{\mathbb{R} }$ such that $g+f_1,g+f_2\in \mathrm{HF}$. We show that this fails for infinitely many functions.