Turns out later that in a VERY INTERESTING TURN OF EVENTS the archers who ambushed and shot him happened to be using arrows from one of the other regions, but the 3rd Prince also happens to have a unit of elite archers in the area. Naturally, everyone accepts this is purely a coincidence.
Basically, it turns out that the Emperor's favorite son is 9th Prince, due to his being the son of his favorite concubine (as in, said concubine has enough influence that the Emperor pretty much listens to her no matter what. The example they use is that if she says a man is bad, he'll be dismissed on her word alone and if she says to keep a man in power, he'll be kept even if he's actively rebelling). Also turns out the 7th prince is one of the 9th's closest allies (7th Prince's mother died early and he was basically adopted by the same concubine). 7th is officially neutral, though. I believe no one expects him to make a try for the Imperial Throne both because he has no interest in it and because since his mother is dead, he doesn't have any active supporters at court/in the harem beyond the 9th's mother. Which is major, mind you, but not enough for an imperial bid.
As for the princes...
The 1st Prince doesn't have any interest in the throne at all (he's a business man first, and until Aheng gets involved, also secretly infertile so he knows not to bother).
3rd Prince is the one who's setting up a potential coup, but between Aheng screwing with the Feng Family (Feng Jinyuan, the patriarch of the family and a high ranking minister, is one of the 3rd prince's supporters) and the Emperor and 9th working against him, he's been checked a few times so far.
5th Prince is a complete womanizer.
7th is, as mentioned, not bothering for the throne but works with 9th from time to time.
Anyways, the other problem the Feng family has is that Aheng's kinda pissed off at the original owner of her body being exiled and thrown to the wolves by her father along with her mother and little brother. Since she survived when she wasn't supposed to, they're stuck with her being a massive pain in the ass. She's also a major embarrassment to Feng Jinyuan, since he kinda upset a fair amount of custom by going 'oh, no, this other concubine is my main wife now'. He has an 'oh SHIT' moment later on when he realizes the Emperor isn't completely displeased with the Yao family (Aheng's maternal family) since the original incident that lead to the banishment of Aheng and her mother from the Feng household was a Yao doctor failing to treat one of the consorts, but as Aheng points out to him, if the Emperor was TRULY displeased instead of just annoyed, he'd have had said doctor KILLED instead of exiled.
Hasn't come back up yet, but I suspect in the future the Yao family's going to play a bigger part. Probably when the Chen family shows back up.
Point is, Feng Jinyuan's swapping of wives around is noticeable enough that even street kids sing songs making fun of him for it, and until he solves that, the 3rd Prince can't afford to marry Chenyu (both because of the loss of face and because it makes it suspicious that Chenyu is the one with a phoenix fate).
Aheng's also sneaky enough to be turning the family against each other while setting things up for quietly yanking herself and her 3rd sister out of the Feng Family. Helps that by the current point in the story, both the Emperor, the Consort (I can't remember her title. 9th's Mom) and, to a lesser extent, the Queen all decide they like her, to the point that the Emperor formally grants her request to have Aheng's mother divorced from Feng Jinyuan AFTER putting out a decree that Aheng is the Feng Family's main daughter, which is another slap in the face to him.
Also, I'll be at least a little bit fair to Minister Feng.
He might not be a good person or anything, but from the point of view of Game of Politics, he actually has a half-decent reason to be trying to deal with Aheng. One of the main reasons he ended up swapping sides to the 3rd Prince is he was originally one of the Ninth Prince's supporters. When 9th was injured and lost his shot (currently) at being the Imperial Heir, Minister Feng started worrying that his family was in trouble since they no longer had a major backer. IE, the whole bit with the Prince Dingan's household, where the consort and her daughter are perfectly willing to humiliate the current Main Daughter Chenyu in public due to being annoyed at Aheng. If he had a more powerful backer at the time, they would...well, probably have done it anyways, since they aren't the most temperate people, but in general most people would hesitate to mess with his family. The need to swap backers also makes Aheng extremely inconvenient, since she's the fiancee of the 9th Prince. I mean, it's a bad idea and shows he is not that trustworthy, but you can see at least one semi-reasonable (by his point of view) reason to get rid of Aheng beyond 'mwahhahahahhaah i have an evil mustachio'.