Thick spots tend to occur when the twist works its way into fibers you haven't drafted yet, which then won't draft, leaving a lump of thick before you can draft it out thin again.
For even drafting, look at how close to the thicker roving you are letting the twist get during spinning. What you want when learning to spin is to keep the twist more than a staple's length away from the thicker roving that still needs drafting -- that way you can draft out the roving to the even thinness you're looking for.
As you get more experience, you'll find you can actually let the twist get within about 1/2 a staple's length while still being able to draft out the fiber ... your control will increase with practice.
Now, all that said, if thick & thin spots persist, you might try to start spinning from the other end -- some rovings are directional, drafting more easily/smoothly from one end than the other. Also, predrafting the fiber before spinning may help open it up and draft more smoothly during spinning.
Enough that in a year or so, when you're looking to spin thick and thin again, you'll have to rediscover how to grab the thick part of the roving and let twist _into_ it to purposely spin a thicker spot ;-)
Do you have tips for spinning a more even single? Comment here or contact me with your suggestions.