Repair and refurb lawn over winter

Hi all I have a lawn approx 60ft by 10ft.
It's 18months old grown from seed.
In the summer I had lots off cracks that I filled with top soil.
I've kept it well trimmed.
watered through summer.
Today I've given it its final cut if 2020
I'm in Essex UK.
I'm planning on doing a hand weed and air holes. Next weekend
3/4 lawn is quite nice last bits mainly weeds.
Over the months the soil has settled and I have some ups and downs, ect.
One edge was bald, I've cut from healthy edge and replanted to other side.
I'm looking best way to say kill off the weeds and level over winter so I can seed and fertilize in spring.
I'll also be erecting a fence to prevent access to as much as possible though cats and wildlife is impossible lol.
Suggestions please. I think I done pretty well so far . Never having a garden before. I'll add photos Tuesday as it's dark now.
Ps grass seed was a mix or donations.
My weeds are low level ones. Mainly 3 different types. Clover,

 

Air holes are a good idea, then stay off it while it is wet. I rather like clover in a lawn , I’d happily take clover over my mossy lawn!

 

It looks quite healthy no dead few dips board patches. Bigger weeds removed by hand. Will air it this afternoon

 

can anyone identify these please

 

Nicely airyated how ever it's spelt.
What best UK product to use this time off year to kill off those weeds. Please.

 

If you've got the patience, you can dig out each plant. Try to get the root a well. That'll work best for the buttercups (rounded ragged-edged leaves).

The other one looks like bugle. Again, if you can dig it out then that's best. It's more of a clump though, so it could be tricky. TBH you might be best giving it a very tight spray of more general weedkiller and accepting a little loss of grass around.

Spray-on weedkiller kills weeds well, but it does also kill the grass too. A good trick though is to get a fairly small paintbrush, pour some weedkiller in a pot, and carefully paint it onto the weed leaves. Be careful not to spill any, of course. (Putting the pot on a drip tray is a good move.) You want to check the weather forecast to make sure it's not going to rain at all for at least a day too.

 

You can use a spot spray lawn weedkiller, this kills broad leaf plants but does not effect the grass.
Not sure how effective these are out of the growing season.
Also you can get a granular Autumn feeds, again not to sure how effective on weeds they are.

Doff Lawn Weedkiller 1ltr Spray Ready to Use 5013655005301 | eBay

 

Way too late to spray now due to temperatures.

herbicides should be used when the plants are in full growth so spraying chemicals or applying them when performance is projected to be low is not great.

its worth noting that if the grass plant is weak the herbicide can damage the plant.