To take benefit of the best prices you will need to prove you are not a business traveller, en route to the airport, so you will need to give either your flight details or your holiday booking reference. I don't think you will need to provide this when you arrive at clacton-on-sea station but you may be asked to provide them on your journey from clacton-on-sea station to the airport.
Going to the airport on the train is great, you can take a nap, read a book (even at night), play travel scrabble, or just gaze out of the window... The other restriction to buying these cheap tickets from clacton-on-sea station to heathrow airport is that there must be at least two grown ups travelling together. But another great advantage is that children go free. All children under 5 go free, and then one child over 5 but under sixteen per grown up travelling on the train to the airport.
make sure you leave enough time to get to heathrow airport early enough to eat, shop etc, so check the times of trains from the station before you go. If you end up with time to kill at heathrow airport you can always read a book or play travel scrabble again! As you have bought your ticket in advance, you don't need to worry about the opening times of the station.
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