Thinking of Roman gladiatorial combats, your cyclists would be allowed - no, expected - to use weapons,
probably spears, javelins and the like.
Free women could have the advantage of protective clothing, but the more armour the more weight of course
(any clothes will be hot and sweaty under the blazing sun),
and nimble slaves could use nets or lassoos, like the naked reticularae fighting heavily-armed gladiators.
Slaves would be chained to their mounts, if they crash they'll be dragged along on the sand and ripped to ribbons
(cf chariot race in Ben Hur).
Wild beasts - wolves, lions, mad bulls etc. - released into the arena during the race would liven things up!