I've seen the position of Second Officer being advertised for some short-haul airlines, but I don't entirely understand what roles and responsibilities they have on the flight deck.
I know a second officer exists on long-haul flight, which allows for crew to take rest breaks through the duration of the flight. So what do they do on short-haul flights? Are they still under line training?
Definition
A second officer (SO) is third in line of command on an airplane. The purpose of a second officer is:
a Second Officer is used on international or long haul flights where more than two crew are required to allow for adequate crew rest periods.
I have never seen anyone besides a captain and first officer on a short-haul flight. Such flights normally don't even require either the captain or first officer to take a break to rest.
Duties
Second officers are qualified pilots and also type-rated for the airplane they are on, similar to the captain and first officer, but only with lesser experience. I do not know the exact career path for them, but most probably they are promoted to become first officers. This is same as first officers are promoted to become captains.
Jobs
There are certainly jobs available for second officers, but only on airlines which operate long-haul flights. Regional airlines with flights only for a 2-3 hours don't require a second officer1.
Cathy Pacific is hiring (also here). The experience of a second officer on Cathy can be read here. Aviation Job Search is listing many positions.
SO on Short-Haul
Each airline can have its own definition of second officer. What I wrote above is the general meaning. You mentioned JetA and Thomson (in comments to the deleted answer) but I couldn't find jobs for SO.
Flight Engineer
Their role is different than pilots and now they are a diminishing breed. Most of them were not pilots. Again, an airline may label a flight engineer as second officer, though Wikipedia did not mention that.
1: The aftermath of Germanwings crash can make some changes in regulations.
