The worms are carrot fly larvae. I haven't grown carrots in a long time but these flies are a notorious pest. There is some advice here -
http://www.gardening-naturally.com/acatalog/Cabbage_root_fly_and_carrot_fly_control.html

Moongrower is the go-to expert on here re veg I reckon. I hope she can help.
Companion planting and low fencing seem to be the way to go.

We planted our carrots between two rows of onions this year and even covered with fleece to start with. The root fly still found them... Next year we will be trying the garlic spray on them the minute they emerge from the ground.

The other thing I recently discovered is it is better to sow the seed very, very thinly and not not thin out the carrots as they are growing and if you do have to thin do so when it is wet. This makes sense as the carrot fly can smell the crushed foliage a mile away!

I'm thinking of mixing the seed with sand next year to see if that helps. You can also buy carrot seed on a tape (at least I 'think' you can) so if it is only a small quantity that might help. In my experience even the seed that is supposedly resistant to fly gets some damage.

At the end of the day unless you are spraying with chemicals, which we wont, it is almost impossible to have carrots with no fly damage at all.

Thank you Beattie I don't really feel like an expert... just years of growing and trial and error :-)

Every year, I get worm in my carrots starting late August and early Sept. in Prince Edward Island, Canada. Last year I bought a carrot type that was resistant to carrot wormfrom Vesey's Seed, York, P.E. but it made no difference. Full of worms. I can not harvest them earlier because they are not big enough.
I heard that a moth lays eggs on the greens and they hatch and the worm goes down in the ground. Someone told to put a fine netting over the carrots in August to prevent this from happening.
Also can these worms stay in the ground over winter? We do have heavy frost.
I am truly getting fed- up with this problem. I love carrots so much but I am beginning to think it is useless.

Can you guide me in anyway, please.
Paul