With these troops? My safe and sound advise is to:

RUN AWAY!!!!
That might be my best bet :) Thank you Cheetah and Gray Beard for your profound insights. Last night I had a few skirmishes with the Mongols - I sent out my Master Of The Horse with many Skythikon, Byz. Lancers, Alans and a few Khawaratsyournameians. By sheer luck I caught an isolated and depleted half stack with the Khan himself - and managed to capture and ransom him TWICE in the same year!

That filled the Empires war chest with ~40.000 florins, most of which was spent on recruiting. I have bitter need of these soldiers, as the Horde has found a way around the bridge east of Antioch and is now besieging the city itself. Luckily, Antioch is very well garrissoned with 10 Byz. Spearmen, 4 Dismounted Byz. Lancers, 4 Trebizond Archers, some Alans and Basileus Ioannis The 8-Chivalry Wonder. Tonight the hammer strike will fall - I hope the large stone walls and ballista towers of Antioch will hold.

You recommended a quick sally to destroy the Mongol siege equipment - is that really an option, considering their large cavalry and missile force? I fear that my two cavalry units will be ripped to shred before they can even reach the trebuchets.