Help finding PWM IC with marking on device 'FA=606'. Package of device is SOT23 - 6
TL;DR; It's likely a Richtek R7735GGE PWM flyback controller.

There are many controllers in this format all with the same package (SOT23-6) and similar pinouts, so it's hard to narrow down the exact ones. So a bit of sleuthing was in order.
Most of them seem to use a similar format for their markings, something along the lines of ###YWW where ### is some short code for the series, sometimes two, sometimes three characters, Y is the year code, and WW is the week code in the year of manufacture.
So what we are looking for is something with a part code beginning with FA, and possibly FA= but that seems a bit odd for a part number. A bit of searching led me to this rather old document, which gives FA but for a completely different form factor which is a bit upsetting. However what it does show is part numbers that are ??= - the very strange equals sign, which seemed like too much of a coincidence to ignore.
Some hunting around for Richtek SOT32-6 PWM controller images led me to this:
Image Source
Aha, we have a part marking of |FA=S03 which is the same strange part number, but with a different year code. I've including the questionable 1 character which looks more like a | on this image, and may well be simply a pin 1 end marker (sometimes packages are marked with a line across the end which has pin 1 rather than a dot).
That gives us RICHPOWER (a.k.a. Richtek) R7735GGE. Now the full datasheet for that doesn't list the marking, it just says contact your local distributer, so I can't directly confirm my hunch, but it seems pretty solid from the anecdotal evidence.
From a pinout standpoint we have below, which matches sufficiently with the names in your comment:
1 - GND
2 - COMP (voltage feedback from opto coupler)
3 - PRO (optional external protection for overtemp or voltage)
4 - CS (current sense)
5 - VDD
6 - GATE (output to external FET)