It all depends on how large your forces are in Georgia and (God forbid) Khazar. In my last Polish campaign Horde's starting forces amounted to app. 5000 troops. Playing Turks, if you can spare around 3000 men to counter them in Georgia, why not give it a shot? There is a fair chance they will go towards Poland, Livonia etc. if you maintain heavy garrisons in Georgia.
If, however you have taken many lands in the east and are spread thin in that part of the map - pack up and retreat to Georgia where you can hope to make a good stand.

Tactically they are not an easy enemy to fight against. Your HA, no matter how good, are not a good answer to the Horde because Mongols are highly mobile on the battlefield and have LOTS of Mongol Warriors (foot archers with very decent melee stats and high discipline albeit no AP attack). Toe to toe any cavalry units you might have are not a smallest patch on Mongol Heavy Cav. I have seen my Boyars backstabbing those guys, while they were being pinned down by AS, and still get creamed. Funny, huh?
I generally advise lots of armoured spears, some fast flanking units to wrap around their forces and some decent valour AP foot skirmishers to shoot anyone in the melee. And have some reserves handy.
Expect heavy casaulties.

In my experience dealing with Mongols is a lot easier when in command of Catholic armies (heavy armour, crossbows, lots of spears). I am by no means great at tactics however so you're welcome to whoop their backsides with Turks in my stead.

Good luck!