What you are looking seems to be the vertical path of a star in the Hertzsprung-Russel diagram (HRD).
The only problem is that stellar evolution is pretty complicated. See here a few simulated trajectories for different masses and chemical compositions from Bertelli et al. 2008
Especially before and after the main sequence (i.e., regular boring hydrogen burning) the luminosity evolution becomes extremely variable.
From the simulations above one can also derive respective age-luminosity relations (taken from Danchi 2013):
Again, you can see that the relation is very complicate for young or old stars and is fairly constant during the main sequence. The problem is, that it changes too little during most of the main sequence. So for most stars even with good mass and metallic measurement you won't be able to accurately estimate age just from luminosity. In addition to that, there are still uncertainties in our models.
Another thing one could do is trying to improve the mass-luminosity-relation by including age. I think this is also what the title of your question implies. The problem here it simply that we generally don't know the age of a star. But if we do, as Rob Jeffries explained in his answer, it is in principle possible.
One think that is done to get age from luminosity is to go all in and calculate so called isochrones in the HRD. These are lines with stars of same age but varying masses and metallicities and can be derived from simulations. If one then measures the luminosity, temperature and metallic one can look on which isochrone the star falls in the HRD (and therefore what age it has). This however is still pretty inaccurate, especially on the main sequence, and is mostly done with whole star cluster, where statistics makes things easier.
This is however not my field of expertise so I would be happy it an actual expert could chime in. :)