This sounds like an AV \ main board failure, which would be around £150 in parts.
Often you find the circuit board components start to fail and as they warm up they start to make connection again on the board and hence the TV starts working, this type of fault will ultimately result in the display not coming on at all if left.
If you would like me to recommend a suitable repair centre in your area please provide me with the model number of your Sony TV, and your postcode.
Good evening, I havent heard from you, I just wanted to follow up to see if you require any further assistance?
Ok, appreciate if you can rate the service received so I may close the case.