Does God forgive sins that we continue to repeat?

The short answer---YES.
The long answer.He forgives you based on His perfection.
We have to listen, again, to the wisdom of Tim Keller.
It is based on His perfection and what His perfection has done for us.
After 7 times, we wonder.
If you repented deeply the first time, then you would not do it again.
Almost never is our deepest repentance enough.
He gives blessings based on the purity of Christ's work.
If repentance is partial, spotty, then he will forgive you.
However, what you may be concerned with, not whether you have lost your relationship with God, but you may be concerned about what we are doing wrong.
Generally speaking when we repent, and we do not change, but there is some attitude that we are not getting at.
If I fib, tell a lie, it is never fudged but it is due to someone being unhappy with us. The lie is coming from an idol. The lie is coming from peoples approval.
I have to get underneath the far idol, not the near one.
God will not throw you away. I realize that my repentance is not cleaning me out as much as it should.
If your forgiveness was based, in the first case, not on you, but on Jesus, why do you think that would change?