A few figures-the battery charger reads 43.7v unplugged .While charging the battery I saw some cells were 4.4v during the charge and believed that this was damaging to the cells so stopped charging at that point. So is the 4.2v maximum voltage per cell to apply only to a fully charged cell while not on charge?
4.2v is the highest the cell should see. Sometimes a BMS will allow another few hundredths of a volt above that before it hits HVC, but not usually two tenths or more.
If the pack is not turning off charge on it's own, it's BMS is either defective, not present, or bypassed, or is being charged from the discharge port instead of the charge port. (or it is so badly designed that it qualifies as defective)
When I first connect the charger the battery voltage is 39.5v and after 10 minutes of charging the battery reads 40v and the pack is warm to the touch. Does this sound OK?
If the cells are heating up noticeably during charge, then it's likely the charge rate is too high for them, either because they are not designed to charge that fast or because they are aged so much that their resistance has increased (which also causes problems during discharge).
If they have a low charge rate, they may also have a low discharge rate, and may be being used beyond their abilities there, too.
The battery consist of a bank of 9 groups of 5 cells each in series. Would this be described as a 9S5?
If there were only 5 cells in series, it would only be 5 x the cell voltage for battery voltage total.
If instead you mean there are 9 cells in series, then that's 9 x the cell votlage for battery voltage total.
Normally there are 10 cells in series for "36v" batteries, so 10s. If yours only has 9, then a 43.7v charger can seriously overcharge the pack. Even for a 10s pack, it should only be 42v.
A 9s pack should be maximum 4.2v x 9 = 37.8v full, so it's charger should be no higher voltage than that.
If a pack's BMS is working correctly, it will stop charge as soon as any cell reaches full. But if it is not working correctly or not designed properly or you don't have a BMS, or you have bypassed it by charging thru the output (controller) port, etc., then any charger that can go higher than the full pack voltage will overcharge the pack.
If a pack is not made of matched identical cells that are all at the same state of charge SoC (voltage), then it will be "unbalanced" and without a functioning BMS a higher-voltage charger than it should have would make it even worse, and have even greater risk of catastrophic cell damage (and potential for fire).
Most packs are not made of matched cells, so most packs require a functioning BMS with a balancing function to prevent imbalance and assorted problems (including catastrophic failures).
Many BMS do not have a balancing function, so while they will still stop charge when a cell (the worst one) gets full, they won't drain off that charge so the rest of the pack can continue charging and fill up the better cells. Eventually the pack will get farther and farther out of balance until there is very little usable charge in the worst cells.
A balancing BMS will help an unmatched (or aging, etc) pack continue to be useful for longer (usually a lot longer).