when can a chess player capture with a king?

Never. In that diagram, the king is in check. It must capture the queen or retreat.
If white couldn't capture the queen or retreat, the game would be over. That would be checkmate.
Never. In that diagram, the king must capture the queen or retreat.
when can't a king capture a coming chessman

Never. In that diagram, the king must capture the queen or retreat.
when can't a king capture a coming chessman
If the attacking piece is protected by another piece, the king can't capture it.
Never. In that diagram, the king must capture the queen or retreat.
when can't a king capture a coming chessman
If the attacking piece is protected by another piece, the king can't capture it.
great, I finally got the answer

Never. In that diagram, the king must capture the queen or retreat.
when can't a king capture a coming chessman
If the attacking piece is protected by another piece, the king can't capture it.
Unless you have more than one king or the king is not the royal piece