If charges are being pressed stemming from an altercation, and the city prosecutors are now trying to prosecute, is it possible for the accusing party to change their mind and get them to drop the charges?
Yes, it is possible for an accusing party to "change their mind." It is also possible for prosecutors to dismiss charges based on this change-of-mind.
If the "altercation" was a domestic violence altercation, it would not be all that unusual for the accusing party to recant their story of abuse and request that the prosecutor dismiss the charge. However, prosecutors rarely will dismiss this type of charge, especially when there is evidence of abuse enough to support it.
On the other hand, if the altercation was not one of domestic violence and there is evidence suggesting the accusation was merely an invention of the accusing party's mind, the prosecutor will consider the accusing party's recantation and may not pursue a legal action against the (seemingly falsely) accused.