I have just a three wire feed to my main breaker box in my mobile home from company meter base on power pole.
That became illegal when NEC 2008 was adopted in your state. (2008-2012 time frame).
There is a bare ground wire coming down from top of pole into the ground I assume, it connects to a rod underground.
That ground rod belongs to the power company and grounds out their transformer to dissipate lightning strikes on the pole top wires. I wouldn't use it.
There is no, ground wire coming out of meter base going to a ground rod.
My main/sub panel in my home has no bare ground wire bonded to the box and all, neutral and equipment ground wires are also tied together on the same busbar which has it's jumper connected to the metal box.
So neither the meter-main nor the trailer has a ground rod. Not good at all.
What that ground rod does for you is arrest surges, spikes and lightning, and makes sure that the actual ground around your house stays reasonably near the voltage of neutral. Which in turn means the voltage of your two hot wires are not much more than 120V from neutral. That reduces the severity of shocks and a
"hot skin" condition. It also helps GFCIs do a better job protecting you for outdoor activities.
The ground wire back to the meter-main does something else for you. It protects you from a Lost Neutral condition where the neutral wire breaks and suddenly the neutral on your home is not at 0V anymore. If ground is bonded to neutral that creates a "hot skin" condition because chassis voltage is bouncing around with the neutral. With separate neutral and ground, the wires are at least separated so one doesn't cause the other, but it also provides vastly better protection if one breaks and not the other.
I assume, I should remove the buss jumper from the box, mount another ground buss to the metal box and move all the bare grounds to it?
That would be useful if you also did all the modernizations - ground wire to the meter and also ground rod to the earth. Otherwise it could make things worse.
I mean, adding a ground bar and separating all your grounds is GREAT. However, leave a neutral-ground bond here in this panel,
until you're able to get
both a local grounding rod
and a ground wire back to the meter-main. You need a functioning N-G bond that has better than dirt between them.
You need the neutral-ground bond to assure that shorts cause breaker trips. Without the N-G bond, fault current would go down your ground rod, through the dirt, and up the electric company's ground rod back to neutral. That would not flow enough current to trip the breaker, leaving it to burn, arc or whatever it's doing. Meanwhile the ground around your home would be electrified, creating a
reverse hot-skin condition!
So you see, the ground rod and ground wire do different jobs.
Should I also drive a ground rod outside and run a # 08 gauge ground wire from the rod to the newly installed ground buss?
You definitely should do that. But you should not remove the neutral-ground bonding if a ground rod is all you have.
The right answer is both a ground rod and a separate ground wire, but the latter is not urgent enough to demand retrofit, as far as NEC is concerned. Still a good thing though.
The usual mobile-home wire is 2-2-2-4 MH feeder, which you note has 4 wires. SE and SER cable are OK outside per Jim Port.