Aside from the mice (and weasels) in clothes using weapons and tools, Mouse Guard is a game that's fairly well-grounded in reality. If we wanted to tell stories with more magical elements to them, how could we go about doing that?
The Arcanist's Dilemma
It's worth looking at another game in the engine, Torchbearer, for inspiration here. Torchbearer is a more conventional fantasy adventure system, with a much different pace of gameplay but similar character creation, progression, and skill systems to Mouse Guard. There are two main obstacles to make it work in Mouse Guard, though.
The first obstacle is that the magic-user (arcanist or ritualist; I'll be using "arcanist" as the general term) is helpless. In the sense that the relevant skills, arcanist/loremaster and ritualist/theologian, are kind of exclusive to the caster archetypes, and neither help nor are helped by most of the other skills in the game. (The "loremouse" equivalent is "scholar".) In a mixed party the arcanist is often acting without the help bonus from other characters, which can be considerable. There's no reason to believe magic would suddenly help or be helped by more skills in the skill web in Mouse Guard, and there's the additional problem of figuring out where in character creation a magic skill should be available. Is it something the Guard trains in, something a town can specialize in?
The second obstacle is the arcanist's scope of play. The game doesn't proceed the same way as Mouse Guard, with a GM's turn with a couple tasks to handle, then a player's turn. In the dungeon, the GM's turn continues on until the adventurers find or create a safe place to camp, and the longer it runs, the more conditions adventurers rack up from stress. The main advantage to arcanist spells in Torchbearer is not that they automatically work - you still need to succeed at a magic skill test - but that they're ways of addressing problems without advancing the game clock. That benefit just wouldn't apply if you try to transfer spells and magic difficulty over to Mouse Guard. The scope of arcanist spells is also generally fixed - they're designed to handle situations on a dungeon scale, while a single obstacle overcome by a guard patrol can represent something on that scale, like repairing a small bridge, or something on the world scale, like making a long journey.
There are two ways you can dodge around these and have "magic" exist in the world without being its own system.
The Easy Dodge: "Sufficiently analyzed magic is indistinguishable from science!"
Taking an honest appraisal of it, Mouse Guard science isn't really textbook science. It's more like action movie science, capable of stepping up to fill whatever role the plot demands of it. For pity's sake, mouse science can fend off bears, which are (proportionately) more powerful than Cthulhu. It's easy enough to just widen the aperture on it a bit, such that a Scientist can accomplish with decoctions, fluids, and vapors most of the things an arcanist could accomplish with magic.
Scientist already exists as part of the skill web, and as far as scope goes, well, in play I've rarely seen it in the GM's Turn as a primary intended test. I've most often seen it used during the Players' Turn, to make weapons and supplies for the future obstacles the Guard expect to face. In that sense it's not much different from preparing spells, but you're operating under the same check economy as every other Guard member in order to do it.
The Harder Dodge: "You cannot leave the magic!"
There's already a system in the game where you can take what would be a normal skill check and roll for something different, where you risk draining yourself on a failure, where you can spend game currency to make a stronger roll. I'm talking about Nature, and tapping Nature. Often in a story of woodland creatures having adventures, if someone "has magic", it's not a matter of known formulas and predefined effects. It's much more adaptable, if not very strong.
I've had some success adapting the rules for using and tapping Nature to make a kind of "natural magic". Here's how it works:
Magical Nature: Add another descriptor to your Nature representing the channel for your magic. This can be a feature of your Body (as "Body: strength", "Body: speed", "Body: senses"), an Element (as "Element: fire", "Element: water", "Element: air"), or a concept you hold in your Mind (as "Mind: shadow", "Mind: illusion", "Mind: mirrors")
Magic And You: During character creation, note down the skill where you have a natural talent, and the trait you were born with. When you're performing a task a normal mouse could perform, this task is within your Magical Nature if your inborn trait affects it, positively or negatively, and it's the skill where you have a natural talent, or any of the skills it helps or is helped by.
Pre-Twisted: If you default to or attempt to tap your Magical Nature for a task, the GM may tell you that the means by which you accomplish the task have unwanted side effects. (For example, using Element: Fire Nature to accomplish an Archivist task, or Mind: Shadow Nature to accomplish a Persuader task.) You can choose to accomplish the task normally.
Going Beyond: You may be able to use your Magical Nature to perform feats beyond the capability of any single mouse. This is a Nature task, but for purposes of taxing Nature, Going Beyond is never in your Nature. The obstacles for Going Beyond are calculated as follows:
Load (Body is free): Elements, Mind
Duration: Just for now, just until, as long as I need, as long as I want to
MAR: A few civilians, a guard patrol, a big family (10-20), a clan (dozens), --, a settlement (a hundred), --, a village (hundreds), --, --, a town (a thousand), --, --, --, a city (thousands)
The Load is what sort of thing your magical Nature is. Elements and the Mind can do things the Body cannot, but at a higher cost. The Duration is how long you need to do it. If the task is something momentary, it's "just for now". If you're setting up other Guard to follow up on what you're doing, it's "just until". If it's a single strenuous task that takes consistent effort, it's "as long as I need". Longer than that is "as long as I want to". MAR is Mice Above Replacement, how many normal mice it would take to do what you're trying to do. If time is a factor, this increases the number of mice that would be needed to do the task. Count the blank spaces as +1Ob when counting up, but you can't stop on them.
Helping Beyond: When you Go Beyond, other guardsmice can help with Wises, and with tools and supplies if appropriate, but Magical Nature is idiosyncractic; any mouse who wants to help with Nature must also spend a Fate Point.
Nobody ever tried to go beyond enough for a village or above, so I can't speak to how well those work out in play, but everything below there went alright. Most uses of the Magical Nature were to replace normal skill rolls, and as such they got as much help as the skill web could get and were scoped appropriately to their original action.
Not A Dodge: Actual Magic Systems
...but this is beyond the realm of my experience. You'll have to author your own, maybe taking inspiration from some of the Burning Wheel magic systems you can find in the main game or the Codex. Just keep in mind the two concerns when you go customizing - how is this magic going to get help from the Guard, and is this magic adaptable enough to act on whatever scale the Guard needs it to?