In slapdash manner around Gravesend without wheels (10)
I believe the answer is:
carelessly
'in slapdash manner' is the definition.
(slapdash means careless or rushed)
I cannot understand how the remainder of the clue works.
(Other definitions for carelessly that I've seen before include "Casually" , "sloppily" , "Without concern" , "Without a thought" , "Incautiously" .)