I would like to travel on the route between Cuzco and Lake Titicaca (Puno) but I cannot find availability.
The Andean Explorer train which services this route does not run daily. 
Between November and March:    The train runs on Monday, Wednesday and Saturday
Between April and October:          The train runs on Monday, Wednesday, Friday and Saturday
Please ensure when searching availability that you are looking on an appropriate day.

I want to travel on the Hiram Bingham on a Sunday but I cannot find availability.
The Hiram Bingham train does not run on a Sunday.  If you wish to travel on this train, you will need to travel Monday - Saturday.

How can I contact PeruRail directly?
Cusco Office (Reservations)
Av. Pachacutec - Wanchaq Train Station
Tel:  +51 84 581414
Fax:  51 84 222 114
E-mail: reservas@perurail.com or hirambinghamreservations@perurail.com