Train from Manchester to York

TPE services from Manchester Airport have a special condition meaning that instead of an Advance being valid only on its booked service it is valid on others for up to 3 hours.

11:06 is probably a bit tight to meet comfortably every time.

So I should look for a train maybe 30 minutes later, or an hour?
Also, I would think the TPE good for three hours is the way to go, right?

This is a duplicate post, you already asked this question, and I answered here: https://community.ricksteves.com/travel-forum/scotland/trains-to-from-edinburgh

for the original question.

11:06 would be OK if the plane landed on time but gives no padding. 11:33 would be safer. If you have an Advance ticket from Manchester Airport on a TPE train and the flight is delayed causing you to miss it, you should take your train and flight ticket (or boarding card) to the railway ticket office at the station and they will check the flight arrival time and endorse your train ticket for later travel - do not just get on the later train. Advance tickets always say "only valid on booked......." but TPE give this special allowance to incoming flight passengers. The three hours flexiblity is based on the booked train departure time NOT the scheduled arrival of the flight.

Transpennine Express is the only train company running from Manchester to York, note that many of their trains go beyond York.

First Class seats are wider and have slightly more legroom, you also get drinks (the coffee is instant....) and cookies. Luggage space is also slightly better. At this time of day Standard Class isn't usually too busy but at peak hours it can get overcrowded.

Manchester Airport station can be slightly confusing the first time due to poor management from the day it opened. From the terminal just follow the signs, you go along a covered walkway to a central foyer and then down in the worlds slowest elevator to the ticket office/ exit level. Assuming you don't need to alter the train you go straight to the door to the platforms (there is usually a ticket check here). Check the screens for which platform, these show time and train destination, so 11:06 shows York but 11:33 will show Middlesbrough - the platform number will also have a sector A or B, this is important as theysometimes have two trains in each platform. The A sectors are closest to the entrance, the B sectors start 3 car lengths along the platform. The individual screen hanging over the platform confirms the train but some of these are set at a bad angle and can be overlooked. The TPE trains usually lay over for a while to allow cleaning so it may be there but locked. Local trains may make short stops behind them whilst this is happening, don't panic! The station announcements are usually dreadful.

And the train changes direction at Manchester Piccadilly (main station), so don't worry about that.

For completeness this is the full official working about the condition on the special 'Airport' advance tickets offered by TPE

As a special concession customers with an Airport Advance ticket, with an origin of Manchester Airport or Lpool Airport Bus, if their flight is delayed, will be allowed to travel on the next TransPennine Express service to their destination. This concession is available for up to 3 hours after the departure time of their original service or if the next service to their destination is timed to depart more than 3 hours after the original departure time they may use this service.
Customers wishing to take advantage of this concession must have their tickets endorsed at the station ticket office.
Customers who take advantage of this concession will not have a reserved seat on the service.
Customers may NOT travel on an earlier departure than the time stated on their ticket and any customer doing so should be treated as travelling without a valid ticket.

Note that I consider the walking connection times from the various airport terminals to the railway station quoted by TPE to be rather optimistic.