I think you probably need to grind the cover crop and dig into the soil
so that it could decompose more quickly. Infact if organic matter
decompose too slowly, it can activate anaerobic processes in the soil
that could impair the following crop.
In such way you increase the content in organic matter.
Another thing you need to be carefull to is the C/N ratio: if the
organic matter N content is not enough, the same organic matter can
reduce the N content in the soil little by little during the
humification process.
At a same time you must consider that cover crop and green manure is
often practiced during wintertime, a season in which rain usually wash
away many mineral elements from the soil, so that green manuring during
this season is anyway a good practice to prevent this loss of nutrient
elements from the soil.
I hope this could help,

Luca Fianchini