I believe that is the "filtered" inlet air for the air pump. Part of the crazy emission control that Jaguar/ BL had to go with when the fuel injection planned for did not work out.
The part about it being a 2 plus 2 is OK with me. Having an OTS, and putting the top down on nice spring and fall days is to die for. The rest of the time..... Give me a coupe, and being odd, I LIKE the 2 plus 2s. No, they are not as pretty, but so what?? Can't tell from the drivers seat anyway, and you can actually carry something more bulky than a FED Ex envelop without re arranging your world.
I think you are in a decent price range for a 2 plus 2, only you should be able to find a much better car to start with. A lot of folks think that since plu- perfect early roadsters top 100K at auctions, their shabby old auto 2 plus 2 is worth a mint. After reality sets in, good buys are there to be had. Beware though, people are waking up to that, and the "unloved" E type are slowly gaining in value and popularity. I think the series 2 version is probably the best driver car for most folks. The V12s can be $$$ to put right, but they are grrrreat when working properly. The S2 cars are 4.2, and are good "developed" cars, with a lot of the (what some folks call anyway) Builn it flaws corrected. They have better brakes, bigger radiator, 2 cooling fans... All of which are correctable in earlier cars. The biggest "loss" is the change to open headlights, tail lights even morons can see, and the bigger, not as "dainty" bonnet opening. Just keep looking, and buy the best car you can find. You will be nuch happier in the end.