Winnipeg - Downtown cheap eats?
You're close to The Forks. The food court has a bunch of reasonably priced ethnic options. The rotis are pretty good, and Tall Grass Prairie is good for pastries, cinnamon buns, breads, and sandwiches too I think. Their banana chocolate bread pudding is really good. The Currant at the forks is pricey for lunch and dinner, but the breakfasts any other day but Sunday are good, huge, and reasonably priced. Sadly, The Fyxx on Broadway seems to have closed, or I'd recommend it for soup and sandwiches. Osborne Village has some good options too. It's a bit of a walk (maybe half an hour from the train station), but there are a lot more choices. Stella's bakery is good for breakfast and lunch. The Second Cup is good for coffee and a treat. Bucaccino's might not be too bad price wise either - not $10, but probably more like $25 for a dinner entree; not sure what lunch runs. Baked Expectations is good too (burgers, sandwiches, and house made desserts). My favourite Chinese place is there too - Spicy Noodle House. I love their dry Szechuan beef, honey spicy chicken, and house beef. Order one main and a noodle dish (I love the beef chow fun with black bean sauce and their Singapore style noodles) and that's a great meal. It would be more than $10 though (probably more like $20ish for two dishes). Avoid Carlos and Murphy's unless you want to drink. There is also Unburger which I have not tried but gets good reviews. The other way on Main street has Ivory which has decent buffet Indian food. I don't recall how much it is though. Even though it's a chain, I like the Elephant and Castle too...although again, $10 would be pushing it. There are lots of less expensive ethnic places on Sargent street, but I would not feel good about suggesting you walk there. Hopefully other people who live or work downtown can give you more ideas.
A little belated in replying:
In the exchange district--Underground Cafe for breakfast or lunch (closed for dinner)
Mondragon if you're okay with vegetarian
A little further north of the exchange district is Chinatown--you might be able to get a lunch buffet at Kum Koon Garden for around $10.
Near Main and Broadway--VJ's for a burger
If you go towards St. Boniface, you could try Le Garage but I'm not sure if the under $10 applies.
IMO, it's difficult these days to get decent meals under $10 in Winnipeg. With that budget, you'll likely be doing fast-food type of things (which would include most of the places at the Forks), and I wouldn't expect many quality meals.