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How do I claim an insufficient material draw?

I had a king and a pawn versus lone king, so hit the offer draw button. My opponent moved, I responded, etc, etc. Finally after 6 tries the game a draw.

I don't think you can.
What you can do is offer draw without moving so they have time to accept without being afraid to lose on time. Alternatively you can just quickly stalemate them for a more "natural" finish of the game.

Because it is possible to beat your opponent in this case, it's very specific try to check this situation in training mode.

You cant claim it. Server detects it when it happens. You had a pawn so it clearly was not in sufficient material

My opponent can't win on the board, but he's allowed to run me off the clock? Is this some kind of bad joke?

If the game keeps going that means that technically he can still win.

If it's any consolation when you run out of time and no possible string of moves would lead to you getting mated (like if they only have a king), then it'll be a draw, not a loss for you.
Thus timing out isn't actually a threat for you in this situation when you cant push for a win, timing out is a threat for them though.

Reaching the same position three times is automatically a draw.

Ive never reached the same position three times because girls claim i have insufficient material

#6 Technically incorrect.
In king&pawn vs king. The player with king can only either lose or draw depending on who times out first

In this particular case nothing drastic will happen, draw in case of time-out. But imagine a closed fortress say pawns on a4, c4, e4, g4 with adjacent Black pawns you can do nothing about but to (pre-) move quickly. Speaking of lichess, otb it is immediately draw after formation.

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