I don't know much about flutes.

I just play around with a tin whistle

now and then, but I know the experience of notes that aren't quite where they should be. I tried a program that converts the sound of a voice or a musical instrument into sheet music. What a shock

I got with a tin whistle that isn't supposed to be producing enharmonic notes anywhere they aren't present on a D-major scale whistle, but did again and again and again....
The professor tells you to play long tones because you're going flat. He probably thinks you don't have the lung capacity to push the notes higher consistently while playing longer pieces, and that you're playing tired.

The long tones are a physical exercise to make your lungs stronger and give you a little more comfort

with the instrument during longer musical works. Keep at them.

(Note: If you're going to play the flute, it helps to stay in good, physical shape.) Breath control is important for low notes, and long tones can help to build the related strength and control.
The other problem you may have (which I share) is a lack of perfect pitch.

Otherwise, you'd go for the "fast air" and push the pitch up a tad if you were going flat; its not that hard, even with these dizi's I just acquired, to shift pitch +/- fifty cents.
If you have a tuner, you eventually need to use it with your long tones. Don't just build strength; build awareness of what you have to do with your embouchure (I like that word - makes me sound like I know what I'm talking about

) and where you point the airstream to get the pitch where it needs to be on problem notes.
I don't worry about it myself...too old to become a real musician!! Keep that in mind before you believe a word I've said!
GOOD LUCK!
