How to reduce a problem?
You are thinking in the right direction, and you almost got it right. What we need is a partial computable map $r : \mathbb{N} \to \mathbb{N}$ such that, for all $n \in \mathbb{N}$,
$$n \in \mathrm{HALTEMPTY}_{\mathrm{TM}} \iff r(n) \in \mathrm{HALT}_{\mathrm{TM}}.$$
Your instinct about having to do something with empty tapes is right, but you should follow the definitions. An element of $\mathrm{HALT}_{\mathrm{TM}}$ is a code $\langle M, w\rangle$, which means that rather than "deleteing the tape", you should provide a tape $w$. In this case, it should be the empty tape $w_\mathrm{empty}$. So we may define
$$r(M) = \langle M, w_\mathrm{empty} \rangle.$$
Reduction in the other direction is more complicated, you should try to do it yourself.