If my stronghold has been destroyed and I cannot get influence points from the cities I hold, can I still get the points from the secret mission card if the objectives are met?

in the rule book it read

"the defeated stronghold is then removed from the board, and the defeated baron does not gain any influence during the Upkeep phase of this round."

any being the keyword

how ever that was written before the mission cards existed and proably wasn't written with them in mind. with that said the wording under missions variant bluntly states

"each player checks to see if he is currently fulfilling the criteria listed on his Mission card.  If he is, then the card is worth 2 Influence.  Otherwise, the card is discarded from the game and its owner receives nothing."

in my opinion this wording suggests that the mission cards operate separately from the stronghold rule.  the sentence "Otherwise, the card is discarded..." seems to suggest that these are the only outcomes possible when dealing with the mission cards and makes on mention of the stronghold, imo suggesting that the stronghold's existence in this situation to be irrelevant.

i would give the player the points even if the stronghold was defeated

Well, I would be happy if u were right, but I guess there is no certain way to know  : S

To me, the point is that the mission cards  ARE WORTH 2 influence, so you are not exactly GAINING influence; but of course as you pointed out, the mission cards were not included in the original rules, making this issue a bit confusing...

Anyway, also the sentence "...are worth 2 influence if they are fulfilled at the end of the game, after the influence pool as run out" (with bold letters quoted from the rules) may indicate that the mission cards are checked after the influence phase. But yet again, nothing is certain unless we get a word from the designers...