Catholics cite the marriage at Cana (Jn 2) to justify the intercession of Mary. Not very promising material, to be sure, but the NT doesn't give them much to work with, so they have to squeeze blood from stones.
Catholics treat the Holy Family as plaster saints. Not only was Jesus sinless, but Mary was sinless, too! Yet what we see in the wedding at Cana is a real family. Real families have real friction, real misunderstandings.
Paradoxically, family members, especially blood relatives (parent/child, brothers/sisters), sometimes say blunt things to each other that they wouldn't say to anyone else because the natural bond is nearly unbreakable. They wouldn't say that to a friend or football coach or the boss.
You might expect people who are especially close to be more gentle in how they address each other, but it can be the opposite: because they are so close, they take liberties they wouldn't take with someone whose commitment is not a given. They know they can get away with saying things they wouldn't dare say to someone who's not within that charmed circle. In the Gospels, the natural dynamic between Jesus and his relatives is very realistic. Not hagiographic.
Jesus is chronically misunderstood, even by those who ought to know him best, because he's a very unique individual. Human, but so much more. How do you relate to God Incarnate? It can be aggravating to be constantly misunderstood.
Jesus appears to respond to his mother's implied request with a mild rebuke for imposing on him. She doesn't ask him directly, but intimates speak in shorthand. They know each other so well that they don't need to spell out what they mean. Like standing at the refrigerator, with the door open, exclaiming, "There's no milk!" That's understood to be a demand for someone in charge to go to the store and stock up on milk.
Maybe she knows that Jesus is naturally resourceful. As a widow, she's come to depend on him. He's the eldest son. The man of the house. Or maybe she knows the "family secret". Her son is more than naturally resourceful. Even if this is his first public miracle, perhaps he's done miraculous things behind-closed-doors.
Be that as it may, he addresses her as "woman". Although there's nothing inherently disrespectful about that mode of address, it's normally the way a man would address a woman he's not related to, and not how he'd address his own mother. A cue that Mary needs to observe boundaries. Give Jesus some space, especially as he's about to embark on the mission for which he was born.
In addition, saying "what does this have to do with me" sounds like a politer way of saying, "Mind your own business!"
Now, Catholics have a fanciful theory that "woman" is really a subtle allusion to Mary as the New Eve. Of course, there's nothing in the text or context to justify that inference.
Although he accedes to her request, he resents the arm-twisting. He may end up doing it as a special favor for her, but he signals to her that her implicit demand is presumptuous.
Moreover, there's a request-rebuke–assistance pattern in the Fourth Gospel, so even if Jesus makes an exception for her, he doesn't only make exceptions for her.
So the very passage in question appears to rebuff her for acting as though she has the inside track. He puts some distance between himself and his earthly mother. She has limited claims on him. And that's becoming more limited.
There is, though, the interesting question of how her request apparently makes him take premature action. In context, the thrust of it seems to be as follows: solving the problem takes a miracle. In this setting, with so many witnesses, that will be a public miracle. Indeed, acting now will be his first public miracle. Once he garners a reputation as a miracle-worker, it's like setting a timer. When it starts ticking predetermines when it stops ticking. The first public miracle initiates the countdown to Calvary. She fails to anticipate the long-range consequences of her request, but he is ever-mindful of his destiny.