To apply for Discretionary Leave in the UK, you must submit an application to the Home Office explaining your exceptional or compassionate circumstances, supported by strong evidence such as medical reports, proof of family ties, or documents showing potential risks if you were required to leave. The application should clearly demonstrate why your situation does not fit standard immigration routes but warrants special consideration under human rights or other compelling grounds. You will also need to complete the relevant application form, pay the required fee (unless exempt), and provide biometric information as part of the process.