Grammar problems~~~~****

1 is correct. 2 is wrong.

'local centres" is the subject of the sentence. 'offering music, arts and sports lessons' is a participle phrase to modify 'local centres'. It tells us which kind of local centres has mushroomed. Therefore we use a participle 'offering' instead of a verb 'offer' here.

If you use 'offer', then do not use 'have mushroomed".
As the demand for extra-curricular activities rises, local centres offer more music, arts and sports lessons to the public.

上面的人講得好o岩, 我想補充少少.

句子本身應該是 :
Local centres have mushroomed.

平是的 grammar 書會教 relative clauses
--> Local centres have mushroomed.
--> Those centres offer music,arts and sports lessons.
用 relative clause 結合:
--> Local centres [that offer music,arts and sports lessons] have mushroomed.

好似一樓咁講, 我地可以用 participle phrases 代替 relative clauses
--> Local centres [that offer music...
--> Local centres offering music...

再一個example:
The girl has just stolen a candy bar from the store.
--> The girl who is talking on the phone has just stolen a candy bar from the store.
--> The girl talking on the phone has just...

(不用選我，因第一個答得好好. 只不過我曾遇到你的問題，好希望你明白多點. English is fun!)