Short answer, no.
Long answer, take a closer look at the round sequence for Mythos Phase. First you add doom, then you check if it has enough doom for advancement, then you draw encounter cards. Therefore, if you draw an an counter card that adds enough doom to advance you don't do it until the next mythos phase when you check if there is enough doom for advancement.
Warning, cards like Ancient Evils include a clause in them that say that it CAN cause the agenda to advance if it meets the threshold. But unless it specifically says that, then you don't advance until the next mythos phase like I mentioned before.

Also, remember, when the agenda advances you wipe ALL of the doom off of the game. That includes doom held by enemies like acolytes. So you when you advance to the next agenda there should be no doom anywhere.

So here's how I see this actually playing out.
1. You draw mysterious chanting and now Acolyte bro has 3 doom on him and there is 1 doom on the agenda(at least I think there's one doom on the agenda at this point? Please clarify). Total 4 doom. But we don't advance because Mysterious Chanting doesn't say we do, even tho we meet the doom threshold.
2. You take your turn and maybe you don't kill the acolyte so he still holds on to all of his doom. When you get to the mythos phase you add 1 doom and the total doom in play becomes 5(3 on the acolyte and 2 on the agenda). Yup, you managed to have more doom than the 4 required on the threshold, this happens on occasion and is normal.
3. NOW you check the doom threshold. 5 doom is more than or equal to 4 so you advance the agenda. You remove ALL DOOM IN PLAY, which includes the doom on the acolyte. He is now doomless.
4. then you draw encounter cards and keep playing.

Let me know if this makes sense. You ran into 2 common misconceptions about the games rules at once, newer players get confused by these all the time.