How to harden a sloped grass driveway so that damage does not occur that has some softer spots?
It sounds like that soft area has deep water saturation which generally occurs in a low area during wet season. Because it's apparently not just a surface issue I don't think you're going to have a lot of success by adding sand or any other material to the grass surface. The best way to deal with it is to remove/redirect the water. One way to do this is to take up the sod and regrade that area - a lot of work.
However, I would first try installing a french drain in that area to channel the water away. The trick is to get it deep enough  so that it's below the water saturation level and also making sure that you're not simply transferring the problem to another area. It's difficult to do without being on site but I would proceed as follows:
(1) Dig a small test hole in the wettest part of the lawn to see how deep the saturation extends down into the soil.
(2) Determine the source of the water. Assuming your arrows all point downhill it may be (and I suspect it is) runoff from the driveway.
(3) Decide where you want to channel the water. Obviously, it needs to go downhill.    That will determine the length and direction of the drain. It may be possible to run the drain parallel to the driveway being careful not to compromise the driveway base.
(4) The drain should be be installed to the depth of the lowest water saturation point.
French drains are best at channeling water from above and less so from the side. That's the reason you want to make sure it's deep enough. It then redistributes the water through holes along it's length. Depending on the slope you're dealing with you can leave the drain open on the low end to facilitate drainage but it's not necessary.