Is it possible to increase the transaction fee of a sent transaction not yet in a block? Is it possible to increase transaction fee of a transaction that is already  sent and seems stuck?
Today? No.

There was a feature that was developed referred to as child pays for parent however few miners if any are using this patch.

There have been instances where mining pool operators have been offered a bounty to include a specific transactions, but there is no automated process for that and presumably would require a significant incentive.