" If it is common land, then cycling is prohibited, the registration authorities have the power to stop the cycles"

Says Tony Barnett

F1 34, prohibition of driving a Mechanically propelled vehicles elsewhere than on on steel roads.

(1) subject to the provisions of this section, if without lawful excuse/authority a person drives a mechanically propelled vehicle-

(a) on or upon any common land, moorland or anyother description, not being land forming part of a road or
(b) on any road being a footpath, bridleway or restricted bye-way. he is guilty of an offence.

Nothing prejudices the operation of- section of the M2 193 1925 LPA (rights of the public over common lands and wastes lands) or- any bye-laws applying to the lands, or affects the law of trespass to land or any rights to remedy to which a person may by law to be entitled in respect of any such trespass or in particular confers a right to park a vehicle on any such right.

The definitive map and statement has the same meaning as in part 111 of the M3 wildlife and countryside act 1981 (animal health).

I will submit if the areas of open space/access, then cyclist under legislation are permitted on those bridle-paths and restricted rights of way, subject of course good behavior, this might help you with the other cycle problems, if it is common land, then cycling is prohibited, the registration authorities have the power to stop the cycles.

I think that s 189(1) of the 1988 road traffic act will help you, cycling causing a nuisance/reckless riding etc on roadways.