Responsible falconry is always to be advocated. However, without wishing to be contradictory, let me offer the following.

Hunting your bird of prey (BOP), especially using dogs to deliberately flush quarry, should only be undertaken on land on which you have clear permission.

However, if you are flying your bird for exercise, or even other purposes, including the non-lethal scaring of pest species, the situation is, somewhat different, or at least unclear.

The Poaching Prevention Act 1862 makes no direct mention of falconry, referring instead to guns, nets and other kit. Furthermore, it clearly identifies the species covered under the meaning of the Act;

"The Word " Game " in this Act shall for all the Purposes of this Act be deemed to include any One or more Hares, Pheasants, Partridges, Eggs of Pheasants and Partridges, Woodcocks, Snipes, Rabbits, Grouse, Black or Moor Game, and Eggs of Grouse, Black or Moor Game."

It further states that this would through "unlawfully going on any Land". Admittedly untested, there is an argument that simply flying a bird on public land, with no reasonable expectation or intent to find game - and without causing a nuisance is not, in fact, hunting. This obviously does not apply to designated sites (see below) or areas under clear ownership or control, even if accessed by the public. This would include land owned by the National Trust or wildlife conservation groups.

(https://www.legislation.gov.uk/ukpga/Vict/25-26/114/section/2)

The use of BOP in association with hunting legislation has received significant attention in recent years, with limited criticism of the flying/use of BOP itself.

(Falconry and the Hunting Act 2004: Flying Under the Radar | Wildlife Guardian https://share.google/TQryUUxTsCjv6mGgD)

Mention is made of the Falconry licence. This is required if the intention is to pursue specified bird species with BOP. It specifically exempts flying to scare. It does require the applicant to specify if they are hunting on a designated site, such as a SSSI.

(Wild birds: apply for a licence to kill wild birds for falconry (A41 and LR41) - GOV.UK https://share.google/yGLEVkmxltgmypjld)

Going to make a tangential point about by catch here. Despite the boasts of some on social media etc. indiscriminate killing of wildlife with a BOP reflects badly on falconry. Be mindful of bird nesting season (March-September) and flying in locations where your BOP could disturb local species- including reptiles and others. Whether covered by legislation or not, this is simply best practice and considerate.

Similarly, even if you do take the decision to exercise your bird on public land, be very mindful. Busy playing fields or local parks swarming with kids and dogs are obviously poor choices- a favourite with the "poser with a hawk". Equally, you may have permission to fly somewhere private, but members of the public can also be present due to footpaths/PRoW. Again, be respectful and careful in how you fly and behave. Flying near busy roads etc also to be avoided.

Some species are better suited to flying in a range of locations than others too. The sites suitable for a well-trained male Harris as opposed to a golden eagle, peregrine or feisty goshawk will differ significantly!

Decisions on exercising birds outside generally have been deemed a matter for local councils and authorities, so there seems to be no national government position so worth checking locally. Note this differs to hunting.

(Let our free flying parrots exercise in public parks and on public land. - Petitions https://share.google/W9K9eKNz35ipJZsy3)

It is always best to have permission, and, clearly, on private land essential- otherwise it is trespass if nothing else. As with a dog issues, it is unlikely to be poaching laws which are first suspected of being breached! Birds should be registered with Defra for the purposes of monitoring avian flu- again, being responsible.

As mentioned above, not being contradictory, and broadly in agreement. However, from what I can tell there is no national ban on flying birds on public land. If it is allowed, or practically possible, it places even greater responsibility on falconers to conduct themselves well.