Most often these are some variation on the classic blocking oscillator.
It's not a class C self-excited oscillator, at least not usually. It can approach that, when the load has relatively high capacitance (like for a HV converter), and when operating under low bias. More often, it's saturating fully into class D operation: the transistor latches on (due to positive feedback from a drive winding), and so load current grows over time (due to V = L dI/dt, the transformer primary has significant inductance), and eventually turn-off occurs either due to gain dropping off at high current, or something else forcing it.
When turn-off is caused by saturation of an inductor or transformer, and operation is balanced (push-pull), it's a Royer converter.
When half wave (single transistor), it can turn itself off due to limited hFE of a BJT, or with the help of a current-limiting transistor (BJT or MOSFET). (MOSFETs typically can't be used for self-turn-off because they are capable of much higher peak currents than BJTs for the same efficiency (in terms of Vce(sat) or Rds(on) at same load current), while paying no price in terms of drive current.)
Typically, bias current is controlled, varying the time between pulses, and thus average power output. This can be done through a phototransistor type opto. Sometimes, regulation is done by sensing the negative peak voltage on the feedback winding itself, which can use just a zener (and Rs and Cs), saving the opto, though this also gives poorer regulation (the output flyback voltage doesn't quite match the feedback voltage for several reasons).
For your case, the opto might be something more than a phototransistor type. Not sure what a 393 should be doing there -- phototransistors are nowhere near fast enough to communicate gate drive signals. A 393 can be used as a basic switching controller (wired up cleverly; the main downside is the ton of resistors needed to do that wiring, hence proper controllers are preferable), so that's a possibility if so. Or it could just be used as hysteretic regulator, or fault protection. (A common sight in ATX PSUs is a LM339 for overvoltage protection on all the outputs.)
There should also be a voltage reference somewhere, whether a zener diode, or TL431 or a relative. Perhaps this is the small SMT, or the aforementioned current limiting / turn-off transistor is it.
Tim