This is... Very confused.
You need to give more information before anyone can even make sense of this question. A public defender isn't a prosecutor; they don't have the ability to have anyone arrested, or prosecuted. The defendant is the client of the public defender, and the public defender is bound by professional ethics to act in the best interests of their client. The public defender is bound by attorney-client privilege, and anything that they were to disclose to the prosecutor would be inadmissible, and potentially grounds to see them sanctioned or disbarred.
So I really can't understand any circumstances here where a defense attorney would be pressuring their client to take a bad deal and somehow threatening their client's mother.
Prosecutors don't coerce people, exactly. It's more a case of, "hey, we're pretty sure we can get a conviction if we go to trial, and if we get a conviction on these charges, we're going to ask for the maximum jail time and/or fines. But if you agree to plead guilty to a lesser charge, you'll spend less time in jail." And TBH, the state usually can convict, since most prosecutors don't like taking weak cases to trial, and the system is pretty heavily tilted in favor of prosecution for multiple reasons.
It's not like, "take this deal or we're going to break all your fingers and kill your mom", but it does lay out the risks of taking a case to trial. I don't think that's coercive by most definitions, any more than your boss telling you that you can either get to work, or they're going to fire you is coercive.
But yeah, the sheer volume of cases that goes through the courts requires plea deals, because otherwise it would take decades for a case to go to trial.