I think this should be called the *relatively ample cone* and its closure the *relatively nef cone*.  Yes, this already exists and is sensible.

This dissertation might be somewhat useful (although things are done for smooth $X$ instead of singular $X$).

[UMich dissertation][1]


  [1]: http://deepblue.lib.umich.edu/bitstream/2027.42/60794/1/ofelguei_1.pdf