I'm gonna have to have a Ding-Dong School lecture here --- don't take it personally because not everytone is required to know this stuff --- but you are asking for information on a site that is designed for a Do-It-Ur-Self log and we DO offer advice.
I tend to get somewhat verbose (so I've been told) but it's the teacher in me.
I run classes with a hickory switch (for beatings of unruly students, not cooking marshmallows), and never will I set my stick on "KILL" --- but I DO use the "STUN" position from time to time.
OK --- what I just said is not 100% true --- I have been known to go for the throat --- but that's for the most-recalcitrant people --- of whom you will not be an adherant--- right?
Good. Read on .....
This battery situation of yours has to be fixed because repeatedly discharging a battery below 10.8V is a battery killer and sets it up to freeze at higher and higher outside temperatures. (up to: 32°F and I don't do Cillimeters)
A fully charged battery won't freeze at any temperature where a human can live (as a guide) - although it is technically more complicated than that.
A battery that has been discharged badly, is never going to be trustworthy again --- they never really heal up from the abuse - when in fact, they get weaker even after the first time. That bell in your situation has been already rung.
OK -- that said .. do you know what the static (not being used or being charged) voltage should be -- in a battery at rest?
13.2 Volts is considered a fully charged battery on the shelf, all else being correct, and at: @ 68°F or so.
When the alternator is spinning -
and this assumes the engine is running - a charging system
CAN (I use that word carefully) have a charging Voltage of 14.2 or so --- but that is dependent on many factors. As a rule of thumb, 14.2 Volts is a decent "thumb" metric.
Some things that affect charging voltages:
1. Temperature of the battery (this may be a mathematical extrapolation* OR a true/registered value)
2. Dynamic condition of the battery, its cells, the amount of slurry in the deposit waffles, etc., etc.
3. Alternator output with the following considerations:
A) Condition of drive belt and tensioner
B) If there are any damaged (open or shorted) diodes in the rectifier pack
C) Reporting metrics of the DIODE TRIO
D) Condition of the commutator brush(es)
4. Size, condition and conductivity of the wiring associated with/to/from the alternator and the battery
5. THE GROUNDS and the condition of the same.
Bad grounds account for 90% of the troubles in a vehicle for any/all electrical or electronic systems and subsystems
6. Heavy electrical power drains that affect the perceived charging rate of the alternator.
A) a higher than the designed-in discharge of some accessory or component that is not supposed to be pulling that much power. (Think a mobile ghetto blaster sub-woofer or high-amperage sound system)
B) High-draw auxiliary lighting or power accessories (eg: winches, OR Hella-style landing lights, etc.)
* NOTE --- think} ---> 4.2Liter oil pressure gauge --- uhhuh?
/