Your castle is going to have the same defences as any other castle -- walls, arrow slits, choke points to limit points of entry for the outside army, etc. What is less set is the kind of magical defences that the castle will have, either in the form of enchantments and/or wards on the castle itself or in the integration of magic into what we would already use. The other big question will be if your group of soldiers can use these enhanced defences, even if they themselves lack magical potential.
Oil cauldrons enchanted to refill slowly so that there is always boiling oil to pour on the invaders could be one such enhanced defence. Other magical defensive armaments could be finite, broken, or unpowered, but that might not be known to the people laying siege on the castle.
Likewise bluffing is a strategy. Keeing with the D&D analogy, there is a low-level spell that places an aura on something, but does not actually do anything else. These walls just might appear to be super magic proof, but most of it is a bluff.
Caveat: The Magic Elephant
As you have stated, magic is a thing in your world. As such, it is reasonable to think that there are enchanted walls -- castles will be designed to take magical assault into account. They likely will not be enchanted to repel everything, but they will take common spells used in a siege into account when creating them. And some uncommon spells if the castle is important enough or its owner rich enough. So they should be resistant/immune to spells to shape stone and earth for one.
This is not to say they are unassailable my magic -- certainly flying over the level of the walls and casting a Fireball or two down into the main area of the castle is a plan -- something a 5th or 6th level D&D caster could do. Likewise, if the spells fail, they will just be regular walls.
Why the Siege?
Other than governmental pride, my question is why would they siege a castle in winter? Magic really helps to answer that one -- it won't be a comfortable siege, but spells will take the largest bite off the invading forces by increasing comfort and ensuring that there is enough food and water to go around in the event of sabotage or a shortage.
To use D&D Terms, a 5th level Cleric can do a lot to support a group through food creation, water creation, and healing support for things that happen.
For whatever reason, the government does not value highly the castle that was taken. Or rather, they do not value highly retaking the castle immediately. Choose your reason.
As such, the force that they send are newer soldiers and leaders in training. It is a test to see how they act and react to the harsh conditions of winter, the burdens of command, and the logistics of handling a siege. This does not make it any less of a siege on the castle -- but it does mean that there will likely not be optimal decisions made as the leaders are not experienced veterans. They will still probe defences and try to gather intelligence, but these are not super ninja wizards.
Bonus point is that any mages sent are not there to siege the castle. Instead they are sent to make sure that the siege survives the winter without heavy casualties from the cold, accidental food shortages, and anything random that might cause them problems. Alternatively, it is also training for the mages as well.
Once spring comes, the evaluation is over and a more concerted effort to retake the castle is undertaken.