Okay... understand that some of this is going to be unpopular advice.
I do understand the concept of "limited" amounts of particular foods, because this is my family's reality with several food proteins for which there is some limited tolerance... however... NO offense-- but
please, please, please call it something besides "allergy" if your child
can consume the food in limited quantities. The reason that I ask this of others is that when families like mine say "allergy" we
need for other people to understand that we don't mean lactose intolerance or behavioral problems. We mean possible fatality just from a food TOUCHING the wrong kind of other food. We need constant cooperation from other people just for very basic safety. This is why I no longer refer to my DD's limited tolerance for milk protein as an "allergy" though it technically
is. (She just has a very high threshold dose now.)
My DD is still VERY thin-- VERY-- and small for her age, and this is due to life threatening food allergies that limit her consumption of a lot of high-calorie foods. Her growth curve is delayed-- her physical age is behind her chronological age by over a year at this point, and this is not uncommon for children with milk, soy, or wheat allergy. Those allergens limit a LOT of food.
We have chosen to NOT make food a battle of wills. Ever. We have also chosen not to coax or bribe with food, because that is, ultimately, giving the child a
lever with which to exert control. Most of them will opt to use that lever at some point...
There is some evidence that this kind of thing can lead to disordered eating, because food becomes about a lot of things other than food. If it is a locus of control; that's a red flag.
My DD does not like vegetables. She does not like many whole grains. She cannot eat many foods which are calorie-dense, or naturally high in protein/fat. We let her eat as little as she chooses. Period. We make purchasing decisions, and we make notes/verbal asides about vitamin needs and other nutrition.
We control what (well, somewhat, anyway), she controls 'how much' and that's that. DH and I were both forced to eat food that contained or was contaminated with our allergens as children-- and I have to be honest here, kids know their own bodies better than ANYONE else can. My DD has often been proven right about a sudden/peculiar food refusal. Either she's developed an allergy to it, or something in the processing has shifted and she's refusing it because it is now cross-contaminated. She's pretty much never been wrong there when we've investigated.
Being very thin isn't the worst thing in the world. I say that as someone who has been living with this for over a decade, with a child who weighed less than 20 lbs at a year old, and it got MUCH worse from there. Family history being what it is, we'll take that over Type II diabetes, thanks.
I also suspect that we've become (societally, I mean) conditioned to expect kids to be.... er... "plump." I'm not sure that this is physiologically intended by nature, or at least not for all children.
It distresses us when DD goes for 8-14 hours without eating, or for a day or two without doing more than nibbling, but she really DOES seem otherwise healthy, and she's reasonably active and alert. Her caloric intake varies dramatically over a week or two-- some days probably 2000+, and others, more like 600.
I'll also say that when she was younger, we OFTEN made the mistake of pushing food
too frequently. Why is this bad? Because they NEVER really eat much unless it's a "treat" that they truly love. Why would they? These are very intelligent children-- by two or three, my DD had figured out that she didn't NEED to eat broccoli, because "muffins" would turn up sooner or later. Why not wait for the good stuff? Besides...
this never really gave her a chance to GET good and hungry. She never felt "sated" either, at least unless we offered her unlimited amounts of a "treat" when she didn't have much else to distract her... but we were constantly offering her food, so of course she wasn't HUNGRY.
My tips?
1. Make mealtimes about food, and make them some fixed (age-appropriate) length. You don't have to EAT, but you do have to be present for the meal's duration... your choice, and the environment can't provide innate reward for NOT eating. I realize that this is contrary to the notion of feed while distracted, but as others have noted elsewhere, I'm fairly uneasy about that particular pattern of relating to food... besides, this is going to backfire badly once they discover the joys of socializing with friends on the computer, where they will want to save both hands and all their attention for gaming/chatting. DD eats far LESS if we don't get her away from the computer.
Caveat emptor. Just sayin.
2. Involve kids in food preparation, and make eating only a part of that process.
3. Let go of worrying. Most pediatricians and nutritionists will freely admit that they've never seen a child without underlying medical problems voluntarily starve themselves into any kind of health problems... but if you provide the wrong KIND of food (just because they
will eat "that"), it can result in some vitamin/mineral imbalances/deficiencies. I realize that minimizes the natural parental concern... but... uh... I guess I
am saying that for most kids, this isn't as big a problem as it
seems to us as parents when they are 2-5yo. (I remember this concern, by the way.) It's not just about the calories, though. Important to know.
4. LET. THEM. GET. HUNGRY. This was the hardest thing for DH and I to learn. We had to wait until DD said she was hungry, or let her wait until meals. We had to let her learn what it felt like to be hungry. She really does have excellent self-regulation now. Even in a restaurant setting, she just eats what she wants, and then takes the rest home-- even foods she really loves. (Oh, and secondarily, following your kid around at the playground with a bag of mixed nuts is creating a pretty severe hazard for about one family in fifty, so there is that, too.

) Genetics has most of us human beings adapted for fairly lengthy fasts. Without a metabolic quirk in play, it's okay to NOT eat, even for an extended period of time (a day or two). We now just ask "have you had enough?" and "So what did you eat for {meal}?" to make sure that she is paying attention to her nutritional needs.
In case you're thinking that I must not understand these concerns, my DD at almost 14 is just over 5'2" tall (with parents who are above average height), and weighs about 100 lbs soaking wet. This is a far higher BMI than when she was 4-10 yo. She is allergic to several foods, and actively loathes a fair number, as well. She's not now, nor has she ever been, a "good eater." We tried a lot of things to budge that, and frankly, none of it really worked. Mostly, this was parental problem-solving in search of a problem that the CHILD didn't
see as a problem. I think that most of us here can identify how successful extrinsic leverage tends to be there. LOL.