1
My mother is very good at cook.
My mother is very good at cooking.
When a verb comes after a preposition, use the -ing form of the verb: ‘good at cooking’, ‘interested in learning’, ‘before leaving’.
2
I have many cousins, and I have a very good relationship with they.
I have many cousins, and I have a very good relationship with them.
After a preposition, use an object pronoun: me, you, him, her, it, us or them.
Compare: ‘They are my cousins.’ ‘I have a good relationship with them.’