Destitution is something that can affect individuals at every stage of the asylum process.
Sometimes a refused asylum seeker cannot return to their home country and have no papers to travel anywhere else. However, they are not allowed to work nor do they have access to public funds. Such people can be left destitute for years but they would rather face this and feel safe than return to the place from which they fled. Others are left destitute between refusal and appeal.
People seeking sanctuary also face extreme poverty (support rates are around 50% below the UK poverty line) throughout the asylum process.
Ironically, many are left destitute at the point they get leave to remain while they wait for residents permits, NI number etc. so that they can apply for work / benefits or even access a homeless hostel.