Simon you should be asking your teacher the same questions that you are asking here. He will be the one to help you properly in the short term. It is difficult to comprehend what "airy" means so you may get different answers.
I guess it's the middle C - second finger from thumb left hand.
Yeah, this does tend to sound more breathy than the bottom C or the top C.
Saxophones in some parts of their range have to be 'humoured' to play in tune and also to get a nice sound. It may be, as Justin Chune said, that your reed is a little hard, or you didn't soak it enough beforehand in blood temperature water (but don't overdo it or it will sound soggy! A minute or so should be about right if you are using a reed of the right strength for you).
If the reed is too soft you will have problems getting the highest notes. You want one which gets you the high notes but is soft enough to get the low notes, too.
The position of the reed and the position of the ligature makes a difference, too. You need to position your reed in a good light - quite small adjustments can make a noticeable difference to ease of playing and tone.
It takes some time to build up your lip - embouchure (as the French say!). Put in as much time as you can early on and you will notice the improvement.
It could be that your sax needs a little tweak, too. (Buy a cheap one or take out a mortgage on a dear one, they can still need a tweak, even when new! (See this site: http://www.shwoodwind.co.uk/ and spend plenty of spare time reading it and a lot of your questions about the playability of your instrument may be answered...)
But do raise these queries with your teacher! He/she can tell you if it is your horn or you, and offer solutions.
If it is middle finger C that is not clear, you can reach up and press the small key cup above the B with your first finger as you are playing. If that makes the note more clear, it means your upper stack is out of regulation and needs a bit of adjustment.