I don't think your burner is lighting at all. You can prove that the warm water is just leftover by just letting it run and finding it eventually just cold.

Nobody has mentioned a thermocouple. That's the easiest and most common fault of a gas DHW heater. Google thermocouple replacement. No need to drain the tank. Just turn black knob to off, slide access panel open, re and re thermocouple, then follow instructions to re-start your controller (typically turn black knob to pilot, press and hold and light pilot while new thermocouple heats, turn black knob to on, hear satisfying click and whoomph, close access panel)