A mouthy horse is not trying to bite, but they are trying to communicate that they are not comfortable with the situation. Horses that bite or nip have just been ignored long enough that they feel the need to get louder so that maybe someone will listen to them.
Mouthy horses VERY often belong or have belonged to anxious people. People that rush around and fail to live in the present moment. These riders are on autopilot and it stresses their horse out. Horses feel the anxiety and wonder what they should be worrying about. This tends to make the horse cranky after a while because the worried, rushing, feeling doesn't feel good. They also tend to get slapped or punished for being mouthy so that doesn't feel good either. That means every time their human is around, the horse feels worried, anxious, and misunderstood/disliked/ignored.
This is like being around a person that disregards your concerns and only talks about their problems. They worry about everything, and you never feel relaxed around them. THEN on top of that, they yell at you when you try to chime in and suggest ways to relax or even talk about your problems for once. That person sucks to be around and is a huge irritation every time they disrupt your peace. Horses can't escape these people or lie about having other plans. They just have to deal with them.
The only way to help a mouthy horse, or one that bites or nips, is to find out why. That means slowing down and listening to the horse. Making sure they feel heard and respected. Their concerns may not be yours but they are important to the horse. So if you ignore them, the biting won't get better.
How do you slow down? Simple. Slow down. Everything. Slow down the saddling process. Slow down the bridling. Slow down the riding. Find out what makes the nipping worse. Can you do it differently? Can you be more considerate? Can you not make the girth so tight when you first put the saddle on? Can you gradually tighten it on your way to the arena? Can you slow down the bridling and fix those issues you have been sweeping under the rug for months (or years)? Issues like avoiding the bit or walking off or even biting during the bridling. These issues can easily be fixed if you are willing to slow down and explain things to your horse in a civil manner.
It is not easy to slow down when you are an anxious person. It's not easy to slow down for anyone in this fast-paced world, but your horse needs you to slow down, listen, find a better way, and be "with them" when you are with them.