One should ideally daven Mincha before candle-lighting, but if that is too difficult, they can daven afterwards, if they have in mind not to bring in Shabbat with the candle-lighting.
Here is why:
The Mechaber (OC 263:10) records a debate between the Behag and other Rishonim if candle-lighting brings in the Shabbat. The Behag thinks that once a person lit the Shabbat candles they have accepted upon themselves Shabbat and are no longer allowed to do any work that is prohibited on Shabbat. The other Rishonim disagree. They do not think that mere candle-lighting has the ability to effectuate Shabbat. Rav Ovadya (Yabia Omar 2:16) believes that the Mechaber himself paskens like the second opinion he quoted, that candle-lighting does not intrinsically effectuate Shabbat. That also seems to be the general attitude of Sefaradi poskim; that candle-lighting by itself does not bring in the Shabbat.
While the Mechaber discusses whether one can do work after candle-lighting, the Magen Avraham (ibid. 19) and Mishna Berura (ibid. 43) say that the same would be true for davening Mincha, that once a person lit the Shabbat candles they can no longer daven Mincha, according to the opinion of the Behag, because davening a weekday Mincha, like doing melacha, is incompatible with candle-lighting. Candle-lighting is supposed to proclaim the end of Friday and beginning of Shabbat; davening a Friday afternoon Mincha after one has lit their Shabbat candles contradicts that.
Even though the Mechaber quotes an opinion that candle-lighting intrinsically effectuates Shabbat, he at the same time also records (ibid.) a debate about whether one can prevent the candle-lighting from effectuating Shabbat through a tenai mechanism wherein the person lighting the candles has in mind that they do not want the candle-lighting to effectuate Shabbat.
Following these two debates, the Rema writes that the Ashkenazi custom is that “the woman who lights candles accepts upon herself Shabbat, unless she made an explicit mental tenai to negate it.”
It would therefore seem that if a person lights candles they are no longer able to daven Mincha because Mincha, like doing a melacha, is a weekday activity, and weekday activities are prohibited once a person has brought in the Shabbat by lighting the candles. It is, however, at the same time clear that the candles’ ability to bring in the Shabbat is contingent on the lighter’s volition, and that if they mentally reject that process, it is ineffective – Shabbat does not arrive.
The answer to your question is, therefore, rather straightforward: ideally one should follow the custom suggested by the Rema whereby they bring in the Shabbat by lighting the candles and, in order to avoid categorical conflict, daven Mincha beforehand. But if it is too much of an imposition, one has the option of circumventing the arrival of Shabbat by having in mind when lighting the candles not to accept Shabbat upon themselves. If you do that, you most certainly can daven Mincha after you light the candles.
While according to the Rema one would require a tenai in order to be allowed to do weekday activities after candle lighting, it is important to note two approaches which justify davening Mincha after you lit Shabbat candles, even without making a tenai.
Taken together, these opinions form a strong basis for allowing one to daven Mincha after candle-lighting even without a tenai that negates the process, since, according to them, candle-lighting never fully effectuates the arrival of Shabbat.
While this is theoretically correct, my above suggestion (to light the candles with a qualifying tenai mentally negating the effect of the candle-lighting on Shabbat) is halakhically more appealing, since it satisfies a broader swath of opinions on this topic.
PS. There is also another angle to consider. While the gender-based distinction of the Bach – where women bring in the Shabbat via candle-lighting and men do not – is quoted by innumerable poskim and achronim, few of them clearly articulate a reason for that distinction.
The somewhat implicit reasoning is seemingly based on the factual difference in terms of who usually lights the Shabbat candles. Since women are the ones who usually light the Shabbat candles, because it is “primarily” their obligation, the minhag to bring in the Shabbat via candle-lighting developed around their practice. They accepted Shabbat when lighting the candles. That minhag, however, never developed for the anomalous case of a man lighting the Shabbat candles. (Rambam Hilchot Shabbat (5:3) offers a practical reason why it is “primarily” women’s obligation. The Tur (263), on the other hand, provides a metaphysical explanation.)
While this seems to be the predominant explanation, some prominent achronim suggest a different approach. Rabbi Shalom Kraus z”l (Divrei Shalom (vol. 4 Siman 60)), an important posek from New York, and Rabbi Zundel Kroizer (Ohr HaChama (vol. 1 Siman 263 seif katan 43)), a well-known Israeli posek, both offer a different explanation for the gender-based distinction. They both suggest that the difference is based on the fact that men used to go to shul on Friday night and women stayed home. Since women usually did not to go to shul on Friday night, candle-lighting became for them the only ritual associated with the arrival of Shabbat. Since this is the only ritual they have available for bringing in the Shabbat, it is when they would do so. Men, on the other hand, would most probably be going to shul and have ample opportunity to bring in the Shabbat there, during Kabbalat Shabbat and Ma’ariv. Since they have these other opportunities to bring in the Shabbat, they never developed the minhag to be mekabel Shabbat during candle-lighting.
While this is a unique explanation, it seems quite plausible.
If the Divrei Shalom and Ohr Hachama are correct, it would perhaps mean that the dividing line is not gender based but is instead contingent on the individual’s Friday night practices. If candle-lighting is the only ritual they will practice on Friday night, that is then when they would need to be mekabel Shabbat. On the other hand, if this person, regardless of their gender, is going to go to shul later that evening where there will be other ritual opportunities to bring in the Shabbat, they would not have to bring in the Shabbat when lighting the candles.
For our purposes then, if you were going to go to shul Friday night, according to the explanation of the Divrei Shalom and Ohr HaChama, you would presumably be allowed to daven Mincha after you lit the candles, since the minhag to bring in Shabbat when lighting candles does not apply to you, regardless of your gender.
While the claim seems plausible, I am not ready to actually pasken like that, yet. Their approach seems compelling, but it requires further exploration. As a matter fact, the Divrei Shalom himself does not fully embrace his own innovative explanation. He still seems to believe, his chiddush notwithstanding, that ultimately the dividing line is gender based; that women, since it is “primarily” their obligation, always accept Shabbat during candle-lighting, and men accept it during Kabbalat Shabbat and Ma’ariv. The one caveat he adds is that if there is no woman in the home, in which case the obligation to light candles is exclusively the man’s, he would have to be mekabel Shabbat during candle-lighting.
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