Eat with and share food is a big one. Birds learn from the flock what is actually food, so they know what is safe.
Also try different prep methods. For instance, carrots can be offered in sticks, coins, diced up fine. They can also be offered raw or cooked. Warm vs cold. Try on a flat plate vs a bowl. Offer them separately (some don't like their food to touch!) or all mixed up as a salad to forage through.
Try to puree it into almost baby food!
Try mixing with foods they are eating. You can also try sprouted seeds.
If you are finding it flung about, chances are they are eating it; they are just messy creatures!
Budgies notoriously do not eat much fruit at all. I stopped trying. I give them a small assortment of chopped veggies every day in a shallow dish which fits over the seed bowl. To that I sprinkle millet seed on top. It took just over two weeks and they were soon eating the veggies in addition to the seeds. So in essence the fresh bowl of seeds is not available to them for at least an hour in the morning feed. By do it this way I found out what my guys do not like and do like. They do not like any peppers.
Here are two photos of what my arrangement is for feeding budgies. First photo shows the two bowls apart and second is how they fit together for presentation. I bought the small flat dishes at a goodwill store.
I think birds like a variety of presentations for their food. Sometime the veggies can be put on a skewer. Sometimes you can hand the greens (wet) from the top of the cage with a clip. Mine especially like carrot tops hanging because they eat them and most importantly they swing on them for fun.