Failure of the carrier to lift high enough is usually related to the carrier rocker assembly.
In the older Marlin's this is a lever and wire spring retained by a small screw in a slot in the carrier.
On the very old Marlin's the spring is a flat leaf spring.
In newer models it's a permanently staked plunger and coil spring in the carrier. This type of rocker is not shown on most schematics.
You can find schematics of the 39-A online and on Gun Parts Corp.
First step is to disassemble the carrier from the right receiver by removing the pivot screw in the receiver so the carrier can be lifted out.
For the old style carrier check the rocker and the spring under it for damage or impacted fouling.
For the new style all you can do is depress the rocker plunger to determine if it's stuck or otherwise not under good spring tension.
For both new and old, inspect the right side of the operating lever for any excessive wear, damage, or impacted fouling.
With the rifle disassembled into the two halves you can operate the lever and see how the carrier is operated.
If the old style carrier has problems the rocker and spring can be easily replaced by unscrewing the screw.
New parts are available from Gun Parts Corp and a few other sources.
If the carrier is defective, the only source of a replacement appears to be a replica made by Wisner's.............
http://www.wisnersinc.com/model/marlin-lever-action-rimfire/If you have a new style carrier with the spring loaded plunger, that's going to be a problem since these were not intended to be repaired, just replaced.
Repair might be possible but special tooling will be needed to re-stake the plunger.
If the plunger is stuck, try soaking the assembly in Kroil for a day and work the plunger in and out to pump the Kroil into the plunger.
An ultrasonic cleaner might aid in freeing a fouled plunger.
Here's a schematic that does show the new style plunger type rocker.........
https://www.marlinspares.com/wp-content/uploads/2012/06/39A12-1.bmp