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In John Farnam's defensive handgun courses, he covers some tactics to use in the situations described above.
The tactics have two strategic goals:
1) Separate yourself from/get rid of the panhandler/possible mugger as quickly as possible.
2) "Shape the battlespace" in case the panhandler/mugger escalates things with verbal "tape loops". (A tape-loop is a phrase you have memorized in advance to use in certain situations, so you don't have to think it up on the spot, and it comes across loud, clear, commanding, and polite, not mumbled, weak, or obnoxious (e.g. profanity)).

The panhandler/possible mugger is accosting you at a certain point because that location gives him and advantage and makes you vulnerable. The tactics basically involve moving out of the danger zone as quickly as possible, using bold movement, feints, your tactical light if at night; and using tape loops: "PLEASE GO AWAY! I CAN"T HELP YOU! LEAVE ME ALONE!" Over and over. The tape loops are used for two purposes - one, to verbally disrupt and command the panhandler/mugger, and two, to draw witnesses attention to the scene and show them you are the victim, not the aggressor. That way, if the situation escalates, and you have to pepperspray or shoot, witnesses will tell the cops about the escalating confrontation and that you were trying to get away; not that they didn't notice anything until you yanked out a gun and shot some homeless guy.