Is an adulterous priest still a priest? If a long-married priest is divorced on grounds of adultery, and in fact had been having affairs throughout the marriage, can he still be allowed to minister as a priest?
The short answer to the question in the title is “yes,” a priest is a priest forever, and no amount of sinful behavior on his part can change that.

However, a Latin-rite priest who is married (save in a very limited number of converts from Anglicanism) would presumably have received an indult to leave the clerical state and a dispensation from the impediment to marry (see below).

Hence, such a priest would not be allowed to minister as a priest, save for giving confession in danger of death.

As the Catechism of the Catholic Church (CCC) explains,


  As in the case of Baptism and Confirmation this share in Christ’s office is granted once for all. The sacrament of Holy Orders, like the other two, confers an indelible spiritual character and cannot be repeated or conferred temporarily (No. 1582, emphasis in original).


What this means is that a person who has validly (i.e., really) received one of the degrees of Holy Orders—diaconate, priesthood, or episcopate—cannot possibly lose that degree for any reason whatsoever. Neither the Church nor the deacon, priest, or bishop himself has the ability to remove effects of ordination. However, as the Catechism explains,


  It is true that someone validly ordained can, for grave reasons, be discharged from the obligations and functions linked to ordination, or can be forbidden to exercise them; but he cannot become a layman again in the strict sense, because the character imprinted by ordination is for ever. The vocation and mission received on the day of his ordination mark him permanently (No. 1582, my emphasis).


In the case described by the O.P., presumably the priest in question is either in an irregular situation—that is, he attempted an invalid marriage while still in the clerical state (see Code of Canon Law 1087)—or else he received both a laicization and a dispensation from Canon 1087 in order to marry someone.

In either case, he is still, so to speak, ontologically a priest, no matter how immoral his behavior is. Depending on what the Church has done in his case, he may or may not be juridically in the lay state.

(But the fact that priest who, having received the necessary indult to leave the clerical state and the dispensation from the impediment to marriage, then commits adultery, does not affect his canonical status. He remains, juridically, in the lay state thanks to the indult—however distasteful and immoral are his actions after the fact.)

Note that, in the Eastern Churches, married men can be ordained priests (although they cannot marry after their ordination). There is also a small number of married Latin-rite priests that are converts from Anglicanism, as I mentioned. In such a case, adultery on the part of a priest would certainly be grounds for disciplinary action, although the law does not specify that laicization is necessarily the penalty.