A I don't think you need to specify the site in relation to your operating centre.
Page 68
If you've noticed an error in this article please click here to report it so we can fix it.
Section 92(1) of the Transport Act 1968 describes an operating centre as the base or centre from which the vehicles are, or are intended to be, normally used.
The key is that the site is not the place where your vehicles are normally kept. With regard to the drivers taking the vehicles home, in my opinion they do need to use a tachograph. Article 4 (12) of EEC Regulation No 8820/85 states that the regulations shall not apply to "Carriage by: (12) vehicles used for non commercial carriage of goods for personal use".
Two interpretations come into play. Firstly, "carriage by" is defined in Article 1 of the Regulations: "Carriage by road means any journey made on roads open to the public of a vehicle, whether laden or not, used for the carriage of passengers or goods". The interpretation of "carriage of goods for personal use (nonconunercial)" would be if your driver was taking a load of topsoil home for his front garden then he would not need a tachograph, but I feel in all other circumstances he must use it.