If the battery were the problem, the voltage would drop much more dramatically. You are seeing that half volt drop in battery voltage due to the current draw of the solenoid, so I expect the relay and wiring to signal the solenoid is good. If it didn't drop that half volt, I would be going after the relay.
So here's the deal with solenoids. There are -two- solenoid coils. The relay sends power to both coils on that same single wire connection.
The "pull in" coil is a stronger coil, draws more current and is grounded through the starter motor hot terminal. When you first hit the key, this is the coil that engages the starter drive clutch into the flywheel and closes the connection to send battery power to the electric motor. As soon as this happens, both sides of the pull in coil have equal voltage, so that coil cuts out. This is where the second coil comes into play, called the "hold in " coil. The hold in coil is grounded to the chassis ground, and maintains the clutch position and power to the motor.
What I expect you're experiencing, is what takes most of these starters out, is a worn short motor brush making a crappy connection to the armature. When that happens, the pull in coil loses it's ground path and won't energize properly. The hold in coil still energizes, but is too weak to pull in the solenoid plunger on it's own. A firm tap on the drive motor can dislodge a sticky brush and restore operation temporarily.
You can at this point take off the brush pack, clean and inspect, maybe it's just gunked up. A replacement motor end cover with brushes can be had for about $30. There's some good videos out there showing starter motor teardown and inspection, it's a fairly simple process.
If the firm tap doesn't fix the problem, you could have an open winding in the pull in coil itself, or in the starter motor. If the hold in coil were to have an open winding, the starter would "jackhammer" in and out of the flywheel. Rare but possible.
Your brush pack design may vary, the video shows the conventional radial type brush, but I've seen axial brushes as well.
https://youtu.be/5Z--ytG70Q0
Patrick (because I have more time than money, and not much of either)