It's a Porter Lancastrian silent compressor, I have the same one. Pretty sure the broken bit is just a terminal block, the lids are long gone off my one as well. Yours is the only other one I've seen so I doubt spares are available You will just have to put it back together as best you can so the live terminals are covered up.
Yup definitely the start relay. It's basically a current relay that switches a start winding in When the current in the main winding drops as the motor runs up to speed the relay drops out and disconnects the start winding.
View attachment 136238
PS - ignore the wire labelled "S" the image is from a compressor project I did a few years ago
https://forums.lr4x4.com/topic/9710-fridge-compressors/