Copper regeneration

[MaelStrom]

Alright Im typing up the lab report for a lab and would really appreaciate any help!
In the lab we started with pure copper and cycled it into various forms through reactions.I am writing out all of the equations, but there is one equation which is giving me trouble.
I have aqueous copper chloride(CuCl₂), and the aim is to regenerate pure copper. So we added pure magnesium(Mg). Since Mg is more active, a single replacement reaction occurs, and thus you get pure copper and MgCl₂
Reaction: CuCl₂(aq) + Mg(s) → Cu(s) + MgCl₂(aq)
But in the lab, we added HCl afterwards to help along the process, to convert any unreacted Mg into Copper. My question is how does this work?Meaning, what would the equation be? I thought it might be something like this:
3Mg + Cu + CuCl₂ + O₂ + 4HCl =3MgCl₂ + 2Cu + 2H₂₀

I dont have a large amount of knowledge concerning reactions with acids, so I really need help with this one. Thanks in advance!