A discovery assessment is how HMRC collect tax for years they can no longer open a normal enquiry into. It's a formal decision with strict conditions attached, and those conditions are often where it can be challenged.
A discovery assessment is a decision, so the 30-day appeal window applies. HMRC must have 'discovered' an insufficiency of tax, meet the statutory conditions, and act within the time limit for your behaviour: generally 4 years, 6 for careless and 20 for deliberate. If any of those fail, the assessment can fall away.
HMRC can raise a discovery assessment where they discover that tax has been under-assessed: income left out, or a relief given that shouldn't have been. The discovery must be new in the sense that the officer has come to that conclusion. There must be no settled practice that the return followed.
Because the time limit depends on behaviour, HMRC sometimes allege carelessness or deliberate behaviour in order to reach older years. Whether that behaviour is proven is a common ground of challenge. 4 years from the end of the tax year: the ordinary limit, where there was an error despite reasonable care. 6 years: where the loss of tax was brought about carelessly. 20 years: where the loss was deliberate, or in certain offshore and failure-to-notify cases.
Where you filed a return and HMRC could have opened an enquiry, they can usually only issue a discovery assessment if the loss was careless or deliberate, or if an officer could not reasonably have been aware of the problem from the information you provided. Full, clear disclosure in your return is therefore valuable protection.
Appeal in writing to HMRC within 30 days, giving your grounds. You can then accept an offer of statutory review or take the appeal to the First-tier Tribunal. You can also ask to postpone payment of the disputed tax while the appeal is ongoing. Talking to HMRC about the assessment doesn't stop the 30-day clock.
Generally 4 years, extended to 6 for careless and 20 for deliberate behaviour, measured from the end of the tax year concerned.
You can apply to postpone payment of the tax in dispute. Interest still accrues if the tax turns out to be due.
You can ask HMRC to accept a late appeal. If they refuse, you can ask the tribunal. You will need a reasonable explanation for the delay, so act quickly.