Can I make my own hand-feeding formula for a baby budgie at home?

There are various recipes available online, usually these are based on products such as cereals for human babies, peanut butter, apple sauce and baby food (e.g. https://www.cuteness.com/article/make-homemade-baby-cockatiel-food). I cannot really recommend those, perhaps only as a last resort. If you can get commercial pellets for adult parrots and grind them with water (pre-boiled) to prepare a slurry, that would work, but the commercial formulae are better. Please note that the food should be at 40-41C and prepared fresh for each feeding.

If you think the baby bird is not being fed enough or is being ignored by its parents, you can definitely try adding in some supplemental baby bird food to see if it helps. Most pet stores carry baby bird formula along with syringes, and the packaging should have feeding instructions for your bird's age and weight. Looking up a few videos on feeding baby birds may also help, so you can learn what a full/empty crop looks like to adjust your feeding schedule. If supplemental feeding isn't helping, or you're seeing other issues or signs of illness in addition to the reduced growth, scheduling a vet checkup is always best to rule out any issues such as underlying health or genetic problems that could be causing it.

Budgies will typically eat a portion of their own food, and then regurgitate it to the babies so it can be more easily digested by them. If the parents aren't feeding the babies, there are commercial baby bird foods available that when added to water, mix up into a slurry that is easy for the baby bird to digest (since we can't regurgitate to them!). As the babies grow and start to explore on their own, they will begin to eat the regular adult food as well.

You could but it would not be a balanced diet. A commercial handfeeding formula would provide complete nutrition for your growing chick. You can purchase it at most pet stores or online. If you can't find one, a better recipe can be found online using your preferred search engine. Follow the link for an example. http://www.indianringneck.com/forum/viewtopic.php

Unfortunately, there isn't much you can do at home if the baby is severely injured or you don't have food for it. Trying to see if a local pet store or big box store that carries pet supplies is open may help you find syringes and baby bird food you need, otherwise. In a pinch, adult bird food can be ground down into a mash and then mixed with water and given via a syringe until you can get actual baby bird food. Keeping the babies warm and keeping any wounds clean and dry can help, however if there are severe wounds or the injuries are bad enough the babies can't function properly, seeking out the care of a vet in person ASAP is best (Many are open still for medical emergencies such as this).

If your budgie's parents are still around they will most likely care for them on their own. However if the parents aren't around or aren't caring for the babies properly, you will need to start feeding them yourself. Most pet stores will carry a basic baby bird food which has feeding instructions depending on your bird's age. You'll also need a syringe with which to mix and feed the food. As your bird grows you can start introducing an adult balanced diet to encourage them to begin eating more solid foods.

The diet being offered currently is not appropriate. At 6 weeks old, Sparky should still be getting a hand feeding formula. A link is included below. Feedings may range between 4 and 8 a day depending on the weight and species. I would suggest having the bird physically examined by a veterinarian or an experienced avian vet to confirm good health. Weaning typically takes place between 8 and 12 weeks of age. At this point, transition to a commercial pelleted diet (Roudybush, LaFeber, Harrison's or Pretty Bird). Introduce other foods gradually and offer a small amount of a fortified seed mix. https://www.kaytee.com/learn-care/pet-birds/hand-feeding