Can I reclaim the cost of my adwords if I set up as a sole trader or some other form of business?
Yes, although you don't really "reclaim" anything as such as a sole trader.
Your AdWords invoices are just part of the expenses of running your business - which at the end of the year you simply subtract from your revenue and pay income tax on the result (your profit).
See [customs.gov.uk...] for more details.
James