The L&Y third rail 'Lightweight' cars were built to run through services from the Southport and North Mersey (Aintree) branches of the L&Y third rail suburban network onto the Liverpool Overhead Railway to Dingle via the connection at Seaforth and Litherland station (the conventional L&Y third rail stock was presumably too heavy for the LOR's iron plate viaduct construction). Through running onto the LOR started in 1905, so I imagine the cars were built around the same time.
If you look where the blue arrow is to the left it says 'C WITH HEAD Ry' - that was the connection onto the LOR. Trains would use the line to the junction before Seaforth & Litherland to get to and from the Southport branch. Trains to Aintree would use the crossover just after the curve to get onto the branch through Linacre Raod and Ford, and then take the curve at Sefton Jnction that crossed over the CLC line to get to Aintree station (not 'Racecourse Station', that was a temporary station on an embankment that was only set up for race weekends, and the line was not electrified past the junction to Aintree).
The units had a very short operational life (on their intended route, at least), as the L&Y ceased through running from Aintree in 1908 and from Southport in 1914. Through running from the LOR to Aintree continued only for Grand National race weekends until the LOR ceased operation in 1956, using the LOR's stock.