Tillage radish

I am in zone 5 in southern ontario. I am working with a lawn that is quite dried out by this time of year and is fairly compacted clay with very little nitrogen. I would like to seed with tillage radish and red clover in the next couple of weeks to get some organic matter and N into the soil.
Does anyone know if I can just slit seed the turf so that the radish seeds is covered in a bit of earth? Will it compete ok with the nearly dead grass so that I get the big fat tap roots I'm planting it for? There is about 1/2 an acre to do.

GroundHog Radish should be planted at a 1/4 to 1/2" depth.
So putting any type of radish seed for tillage at this depth should be ok.

From http://www.tillageradish.com/documents/tillage-radish-resource-guide.pdf
Always mix a grass species with Tillage Radish® planted at 4 lbs/acre

So it is common to mix tiller radish with grass.