The Poskim discuss the times when an Avel should not daven, and I have not found anyone who says that Mincha on Erev Shabbos is one of those times (That doesnt mean that the opinion doesn’t exist, it just means I didn’t find it)
. In fact, in the Sefer Pnei Baruch (Siman 35) he quotes the father of the Shl"a HaKadosh who says that on days when Tachnun is not recited (which would presumably be the reason for the minhag not to daven on Erev Shabbos at Mincha, if such a minhag even exists)
it is in fact a bigger Mitzvah for an Aveil to daven for the Amud.
The Mishna Berura too (Siman 581:7) writes that one may daven on a day that Tachnun isn’t recited. In fact the Pnei Baruch also writes there that this that Aveilim do not daven on Shabbos is only a minhag, but there is no Issur!
The time when the Aveilim do not daven is when " LamnaTzeach ...Yaancha Hashem B’Yom Tzara,Tziduk HaDin" is not recited. And even then it is a Machlokes HaPoskim as to which Tefilos are better for an aveil to not be the Shliach Tzibbur.
On Rosh Chodesh some Poskim say that an Avel should not daven at all, while others maintain that only for Halel and Musaf he shouldn’t lead, but Shachris, Maariv and Mincha are OK. Others maintain that for Shachris too he shouldnt be the Ba’al Tefilah. (See Mishna Berura ibid.)
On Chanukah an Aveil may daven Mincha and Maariv, but not Shachris and Halel. (See Mishna Berura 683:1 and 671:44)
There are Poskim who maintain that on Chanuka, even for Mincha and Maariv, an Aveil should not daven for the Amud (Shu"t Rivevos Ephraim Vol. 1 Siman 443 says this was the opinion of Rav Moshe Feinstein Zatzal. See also Shu"t Divrei Moshe Siman 84)
For a final ruling, a Rav should be consulted, of course, or at least the Gabbai of the Shul who may know the "Minhag HaMakom", which should always be followed.
Important Numbers
