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Hello my name is ***** ***** I will assist you. So if i understood you properly water is coming out when you connect the flush hose but not when running the engine, then most probably the impeller was worn out and it would need to be replaced. To replace it you would need to remove the lower unit of the engine.
IF you need further instructions or assistance let me know.
Can you send me a photo of this hole? I am suspecting that this is the hole where the water pressure sensor is attached.