Yes, you can - as in it is possible - but you must consult with your pastor
So now, can you get a Church marriage done for the second marriage??
In my time in the RCIA ministry, I saw any number of cases similiar to what you describe.
Have you spoken to a pastor yet?
If not, get off the internet and go see the pastor.
Each case must be dealt with on its own merits. You need to discuss the details with your pastor, and perhaps the chancery at your diocese.
Example: A couple in our church who had a civil marriage only, both raised Catholic, had what was termed a "short form" process ... but their case and yours may not be identical.
You need to see your local pastor about this. The process you are asking for may amount to a convalidation. (Please review the Q&A at the link, it appears to be related to your question).
The only valid answer to this question is: go and see your pastor.
As Geremia pointed out, confession (the sacrament of penance and reconciliation) will be a part of the process.