先に and 先ほど: acting first and an event a short time ago
Compare a request to put luggage down first with a recent phone call, keeping action order separate from past timing.
Before another action
先に, read さきに, can specify acting before something else. 先に荷物を置いてください。 asks for the luggage to be put down first.
私が先に説明します。 says that the speaker will explain first. 先に can introduce a future action and does not itself report past completion.
A short time before speaking
先ほど, read さきほど, refers to a short time ago. 先ほど電話しました。 says that a phone call was made a short time ago.
先ほどの説明をもう一度聞きたいです。 expresses a wish to hear the recent explanation again. の identifies that earlier explanation.
Action order and a past reference point
先に電話してから、荷物を運びます。 gives the sequence of calling first and then moving the luggage. 先ほど電話しました。 instead locates a completed call in the recent past.
The exact number of minutes implied by 先ほど depends on context. Ask 何分前ですか。 when a precise interval is needed.
Does 私が先に説明します。 report that the speaker finished explaining a short time ago?
Show the answer and why
No. It says that the speaker will explain first, ahead of another action or person. It does not report a completed explanation.