So who’s correct?
In Mark’s recollection of the events, James and John come to Jesus with no mention of their mother. The verses are very specific in that they are asking him personally. Where they say things like “Can we…” “Grant us…” These are the first-person statements from the two brothers.
In Matthew, the mother with her sons James and John comes to Jesus. It is clearly the mother making the request to Jesus on behave of her sons. “She saith unto him, Grant that these my two sons may sit…”

Two possible explanations have been considered throughout the years.
The first being that these were two separate conversations. One where the mother with her sons came and made the request. And on a separate occasion, James and John alone came to Jesus to make a similar request.

The second consideration and the one I would consider more plausible. Where James and John with their mother came to Jesus to make the request. And in that single conversation, both James and John specifically ask Jesus, and their mother specifically asks Jesus. They all came to plead for the same position with Christ in his glory. Mark recorded the words of James and John, and Matthew recorded the words of their mother.

Mark 10:35-38
Because Mark does not specifically say that James and John’s mother was there does not create a contradiction. The basic argument could be that the writer of Mark only heard or recalled James and John making the request to Jesus. This could also be a writer’s choice. The writer of Mark decided that all the people involved were not as important as the request that was made, and more importantly, the response that Jesus gave.

Matthew 20:20-22
Matthew, in my opinion, solves the dilemma.
“Then came to him the mother of Zebedee’s children with her sons…”
It is this statement that disproves that two separate conversations occurred. There is no need for James and John to come by later, or to have come by before, to ask him again to make the same statements in two separate conversations.
It is also this statement that upholds that all of them came together to have a conversation with Jesus. Matthew either chose not to include the statements of James and John. Or he only recalled the mother’s part of the conversation.

This is not a Contradiction. They are both right.
It is eyewitness testimony of the same situation that includes the same response that Jesus made from two points of view. James and John, along with their mother, had a conversation with Jesus, requesting that James and John sit on the right and left hand of Jesus.