Cathay out of the UK do a few sales every now and then - if you didn't want to sign up for their newsletter (which 9 times out of ten is just marketing, and only rarely has a big sale), then you may want to keep an eye on headforpoints, the loyalty website. An example Cathay UK sale:
https://www.headforpoints.com/2025/0...cific-flights/ ; and the general "When there is something on Cathay page:
https://www.headforpoints.com/catego...nd-asia-miles/
But for what you are choosing, the cheapest economy fare on a very specific date you would be best doing a google flight notification to see if there is a sale, and letting google suggest when to book. Honestly I would be surprised if prices went down that much, and if you are really fixed to a certain date I would be trying to buy sooner rather than later.
Personally in this situation I would also be tagging/searching Qatar Airways routes on google flights, as they have 3x flights a day Manchester to Doha, and perhaps 3-4x a day Doha to Bangkok, so they are more likely to have sales. Plus, Qatar often have their "10% off with Visa" type promotions - however many times their promotion codes exclude the cheapest fares, but their sales do include those fares. I would equate CX and QR fairly similiarly on these routes, unless you want to visit Hong Kong for a stopover (great!) or prefer a 15 hours then 3 hours, vs 2x8 hour flights.