Your intended meaning is unclear.
If you mean 'I have no plans for the weekend', then you could say (in BrE or AmE)
I don't have anything on this weekend.
but 'on' here would be an adverb, not a preposition, a part of the phrasal verb 'have on' (= have as a plan), and therefore unconnected with the noun 'weekend'.
The regular preposition for 'weekend' is 'at' in BrE and 'on' in AmE, so that BrE
I often go away at weekends.
would become in AmE
I often go away on weekends.
I trust this answers your query!