Cereals and purees are unnecessary in the first place. The idea that we 'must' start with those is a holdover from when babies were started on solids waaaaaay too young. And stuff with "chunky bits" -- I'm assuming you mean a chunkier puree -- can be confusing to a baby. It's actually one of the common problems when starting with purees... they still expect everything to be 'smooth' and their instinct to chew hasn't been stimulated. So it's quite natural for the chunks to stimulate the gag reflex when they try to swallow these unexpected things, and the gag reflex causes the puking.
It's a fabrication of the baby food industry that you have to move from purees to gradually thicker and chunkier consistencies before ever getting to finger food. Babies who are ready for solids are ready for finger foods.
The puking when gagging itself is not a cause for worry. It's kind of disgusting but it's "normal".
My suggestion would be to stop the purees and start offering her real food. There is of course a chance that she's not ready for food yet -- yes, even at 10 mos. Some babies just aren't ready until even over a year old. If that's the case then you shouldn't really be pushing the purees on her either.
Since she has become accustomed to the purees, she might be uncertain about real food at first. On the other hand, she might take right to it! Many babies are quite clever and reject the purees outright heh... If yours has "never been a good eater" it could be that she just doesn't like the purees and will do fine on real food, or she might just not be ready yet.
As long as she's still nursing lots you don't have to worry about nutrition, so even if she eats nothing she's fine.
I'd suggest trying something not too soft, which she can hold and gnaw on. Not 'hard' like carrots, but more something like a soft pear, or some pasta, or toast, or steamed broccoli trees. Or you could try small bits of food which she can pick up with her pincer grasp. (And do let her feed herself. Spoon-feeding can cause problems for some babies too. She might benefit from having more control over her own feeding.) Don't worry if she gags a bit at first, again that's normal (and gagging is the reflex that protects against choking). If the gagging is constant, though, and doesn't seem to diminish after a few days of trying, then the odds are she's just not ready yet, and the strong gag reflex is her body's way of telling you that.
HTH
Edited to add:
In case it helps your 'confidence' in the safety of offering real foods to an infant, I offer my photo set of DD's eating adventures. The first pics are of her chewing on broccoli at 6mo, and the most recent show a 9mo confidently shoving pizza, watermelon, beets, puffs, etc into her mouth. We also started a spoon just very recently, for yogurt - I load it and she takes it from there!
http://flickr.com/photos/lovecat/set...7600330803313/