Hi I think you always get some bare/thinner patches in a new lawn and they may need reseeding.
Sounds like your mower is damaging it? at this time of year easier to do.
How high is your mower set? it could well be set too low also if the conditions are wet the new lawn will suffer when mowing.
Unless it is dry I would not cut it now, but if you do need to then use a high cut.
Best to keep off it now and if you do need to reseed wait till spring.
Keep off as much as you can especially when its frosted to as this will damage it.
In the spring mow often (on a high cut to start with) as this encourages the grass to grow sideways and thicken up. Feed it in the spring and reseed if you need to.
Dont worry all lawns look a bit like yours to start with and come next spring it will really start to improve.

If you did the first and second cuts with your particular mower after it started growing,you would have ripped some out of the ground. Even on the third cut, the same would have happened. The first and second cuts should be with shears for just that reason, although a good hover mower is a reasonable substitute. Reseed bare areas next Spring - if any are large, cut those areas with shears the first couple of times before using your mower.

If it is a rhizomatous grass, such as bluegrass or creeping fescue, it should repair itself with feeding and good soil. If it is a clumping grass, such as ryegrass, tall fescue, or Chewings fescue, it will need to be re-seeded, to fill in the gaps, though the clumps will spread a little to fill in very small gaps.
As an aid to fill in, I would mow often enough that only 1/3 of the height of the grass is removed--i.e., if mown at 2 inches high, don't let it grow more than 3 inches high before mowing again. This allows the grass to spend more of its energy on covering ground, rather than regrowing lost leaves.