I have a Tiny88 NANO board which I can upload to OK from the IDE installed on my PC (after installing MH-ET LIVE board via the board manager). But I am not able to upload to it from the Cloud version of IDE. There are 2 versions of MH-ET Live boards listed in the drop down board selections but neither of these will allow the program to upload, it just says "error uploading, check if the selected board is currently available".
So is it possible to add more boards to the cloud version? If so how and if not is there any other way of being up to upload from the Tiny88 board.
No, it is not possible. You can only use the fixed set of boards supported by Arduino Cloud (as well as any derivative boards that are functionally equivalent to the supported boards).
There is no support for ATtiny88-based boards in Arduino Cloud so you won't be able to use this particular board with Arduino Cloud.
As you already discovered, you can use the desktop Arduino IDE. If for some reason Arduino IDE doesn't meet your requirements, we can suggest some alternatives.