Whenever you knit in the round, you are going to get a little dogleg of color when going to the next round, because you are knitting a spiral. The way to avoid this is to begin the next color one or two stitches in the current round, before beginning the next round. You are still knitting a spiral, but the color change is less noticeable. You might vary it a bit and begin the color change one or two stitches after beginning the next round. You might also try alternating the colors one or two stitches before beginning the next round.
Try it and see which works out best. You can always pull it
out and try it until it works. Sometimes the color change really isn’t that noticeable. Hope this helps.
Happy Knitting.

when doing color work twist yarn colors over each other, for example when going from red to green, bring green up over red yarn. This will make a nice smooth color transition with no holes.

EDIT: from

“Note: to avoid the “jog” when changing colors at the beginning of a round,
knit the first round with the new color as usual. On the second round, insert right hand needle tip into stitch below the first stitch in the round on your left hand needle. Pick up this lower stitch and slip onto your left hand needle and then knit this with the first stitch already there together. It isn’t completely invisible as it looks like a slipped stitch, but it looks better than a jog and to the less critical eye it will not be noticeable. (The more rows you have per stripe and the smaller your gauge, the better it looks.)”

If you change the colours each time right after your stitch marker you really shouldn’t notice much of a change mark. I’ve just finished knitting a bunch of striped hats for xmas and I did find that as I was knitting it I had a bit of a gap on each round. However, once I took it off the needles, I twisted the ends together and wove them into the inside of the fabric and now you can’t see the colour changes at all. It looks like rings of colour stacked on top of each other.
I’m wondering if you are marking your rounds or if your colour change was more random and that’s what’s causing the issue. I hope this helps you out.
If it’s really stressing you out, then maybe you could try a swatch using different methods to switch your colours.
One other thing is to make sure you twist the ends of your yarn together a little bit as you’re adding the new colour just so that it looks a little smoother while you’re knitting.
My final suggestion would be that if you are still really unhappy with how the colour change looks, perhaps you could try to place the colour change in a less obvious place on the mittens, maybe along the edge?

Sigh, avoiding the color jog is the bane of any knitter. Change your colors in the first stitch of the new round, but when you get to the last knit it in the top of the stitch from the row below, which minimizes the visual impact of the change. This said, most knitters will make the change either at the thumb side, or in the center of the palm, or simply not worry about it. Another option is to make a *fake* seam at both sides of the hand as Elizabeth Zimmerman recommended for her circular sweaters which hides the changes by marking the beginning and mid point of each round and then every other round knitting that stitch in the stitch from the row below. This works nicely unless you are doing other color patterning, like Fair Isle where it would interrupt the pattern. In your case it could become a design element.