You will not get compensation for everyday mistakes, at most an apology. Examples of this include errors in inputting tax return information, incorrect calculations, wrong penalty notices etc.
To get compensation you need serious or persistent errors. A serious error is something that no responsible person acting in good faith and with proper care could reasonably have done. Perhaps wrong advice from the Taxman that you relied upon.
A persistent error is one where the Taxman continued with the mistake after it had been pointed out to him or keeps making the same mistake or made several unconnected mistakes in the any twelve-month period.
The compensation you will get will cover putting the tax position back to where it should have been, reimbursement of expenses or financial loss and a consolatory payment