This what under the sink looks like in another property of mine. Again water is coming out of the spout on the left when water drains from the sink at a fast enough rate. Is this spout again for a dishwasher outlet? Thanks

That's a connection for a dishwsher drain hose.

Harold's bang on. You are teed off,( a brainch tailpiece would have been better, see image) to pick up a dishwasher. Run the discharge hose up into a high loop and secure it with a pipe strap, (see image) and secure the hose with a hose clamp, (see image). Good luck, Tom