If you can only ride once a week, you will certainly improve but as with any thing else practice makes perfect and the more you practice the better you will be faster.
I had a yr and a half when I first started that I only rode once a week, then for several years I rode about 2 times a week. Once in a private lesson and once on my own. Sometimes three times a week if I was lucky.
Since I've gotten my license though I've been able to ride a lot more...more like 5 times a week. And I feel like it's really been an improvement
However money-wise, I wouldn't have been able to have many lessons when I was younger. After 11 yrs old, my instructor kindly let me work off all my lessons and riding on my own times and I've done that ever since to this day. Could you do the same type of working student program thing?