I imagine from the questions that you have asked that you are planning some project that may require a high capacity lithium iron phosphate battery that you would like to build yourself.
The only large capacity battery pack in high numbers production that uses a number of cells in parallel vs only a couple of large cells in parallel is the Tesla. All of the other EV seem to go with the large capacity pouch cells and then only use a couple in parallel. The advantage to the Tesla is that their power to weight ratio is almost double that of everyone else.
The disadvantage has to be safety. Tesla has 104 patents on its battery pack and most of them have to do with safety. Specifically with how to deal with single cells that short or go into thermal runaway. FYI the Tesla S battery pack uses 74 cells in parallel and then 96 in series.
I have read through their patents and none of them deal with any sort of system to repair or remove damaged individual cells. They just make sure that on the rare occasion that a cell does short or heat up, start fire etc., that it is contained to that cell. They do this by separating each cell by a certain distance, using active and passive cooling, using a fuse at each cell etc.
Tesla brags that they can replace an entire battery pack out of a Tesla S in only 90 seconds, but they make no claims about repairing that battery pack. In fact because of all of the safety features like fire proof foam, it takes a couple of hours for a person to get access to the individual cells of the Tesla battery pack and by that time you have ruined the structure of the battery, so it is no longer useable to the car even if you did replace the destroyed individual cell.
So to answer your question, it appears that they put a lot of effort into preventing a shorted cell from destroying the rest of the pack, but otherwise they leave it there and let the rest of the cells in that parallel group take over.
Remember that the Tesla is demanding a very heavy load from its battery, not only pushing the cells to their limits to get a further distance out of the pack (some owners report cell voltages that dip below 3.0 volts) they also demand a high current for the crazy acceleration the Tesla gets.
From reading their patents, Tesla believes that over charging is much more dangerous than over discharging (this is from tests they have done in their labs). Over charging leads to fires and explosions while over discharging tends to speed up capacity loss.
Good luck in your project. Using lithium iron phosphate cells you are already a magnitude safer.
What you propose should work technically and is probably not too hard to do "well enough" as long as you are sensible [tm].
As Peter says, mixing battery chemistries can often have bad outcomes, but as long as you do not violate basic requirements this could be reasonably problem free.
It is not obvious why you would use a LiIon second battery - lead acid will better match the cost/capacity and general cycle lifetime for given use.
If your main battery was say about 50 Ah (less than 1 hour at 50 A due to C rating usually being at 10 hour rate or even lower) and you wanted both batteries to work together throughout the LiIon battery would also need to be 50 Ah or more. That's not a small battery and you are going to need a separate specialist charger to charge it - which is OK if you have the equipment already - but otherwise adds expense and complexity. You may be able to obtain 2V indivual LA cells or a 4V battery (they do exist) but again are faced with the charging issues mentioned below.
A LiIon battery MUST have it's low voltage discharge limited to a safe voltage. It MUST be rated to discharge safely and continuously at say 50A.
Max charging current is liable to be 25A or 50A (or other - varies with manufacturer). You COULD safely enough [tm] charge a single LiIon cell by limiting Imax to at or below rated max value and setting maximum charge voltage to say 4V or slightly less. You can then allow it to charge until 4V is reached and can float it at 4V "safely". This charges to noticeably below max capacity and increases cycle life. You MUST NOT charge it all the way to 4.2V and float it there - battery death happens soonish.
Note that LiIon will have 4.2V (if fully charged) to start and 3V or so fully discharged. (Lower possible but unwise if long cell life is wanted).
Best Answer
Without further details it is difficult to tell.
Anyway, there is a BIG caveat: jump starting a veichle requires very high current spikes, and the cells in the string must be able to withstand it.
If the lead acid batteries you will use are of the starter type (and not of the deep cycle type, as used e.g. in UPSs) they are designed to provide current pulses that are even hundreds times greater than their normal current rating (e.g. a 50A rated battery could provide 1000A pulses for a couple of seconds).
This is a problem with LiIon cells, which usually can't survive such pulses. You might think that putting in series an equally rated 50A LiIon battery would do, but probably that battery could survive just a 200A pulse, and maybe for only a few milliseconds).
Moreover, the internal impedance of the LiIon cells could be greater than that of the Lead Acid battery, so you end up with a pulse of power being dissipated in the LiIon that could raise its temperature above safe limits.
See this comparison table between rechargeable batteries at BatteryUniversity.
TL;DR: you should get the datasheets of both the Lead Acid battery and of the LiIon battery and examine their characteristics. Only then you/we could tell if what you have in mind will be safe to do.
SAFETY WARNING: lead acid batteries are quite rugged and they can withstand even strong overloads for a short time. LiIon CAN'T DO THAT Overloading a LiIon cell could heavily damage it, which in turn could make the battery explode or vent with flames!!! Don't try your experiment without knowing the characteristics of your batteries. You don't merely risk to damage your batteries, you risk your life (especially with big LiIon battery packs)!