If you fear returning to your home country due to a risk of persecution and you wish to claim asylum in the UK then the Home Office do not charge a fee for you to do this.
Once you have claimed asylum then you can make an application for support known as NASS. If you r application for support is accepted then you will receive a weekly payment that you can use to help with buying food or everyday essentials. You can also make an application for accommodation.
If you are granted asylum you will be given 5 years refugee status and at the end of the five years can apply for settlement in the UK. At the present time the Home Office do not charge a fee for this application.