Apply using an agent here. There are no published working numbers for the CAA permit office.
More than one landing takes more work and can take up to six months to get approval (like an operations permit). Once you have this approval can you go through normal channels to arrange a permit.
Single flights can be authorized without going through the six month long process, but these are done on a case-by-case basis and only if relevant to special events or emergencies. American Flight Support will be able to advise on what is possible in your specific case.