My understanding is that initial acceleration from the blast and following solar wind will clear the area around the asteroid for spacecraft navigation (in reasonable time). A 2-3 kilometer M-type asteroid (or smaller) would not have enough gravity to be able to keep around the dust cloud from the blast?
One kiloton moves both quite a lot and probably less than you need.
This study on nuclear craters along with a bunch of maths relevant to OPs interests has a table on page 24 summerising various actual blasts, including some subsurface low yield shots that got crater sizes around 100 feet and moved around 1 million cubic feet of material. While an asteroid will have lower density much lower gravity they provide a useful hard number to start with in terms of just turning blast into moved mass.
Moved into metric that is 28 thousand cubic meters, the table lists alluvial soil which is around 1.5 tonnes per cubic meter so 'moved' around 42,000 tonnes of material. It is notable that the table suggests smaller yields get more cubic volume moved per ton of yield.
Shifting 42k tonnes of material is great, but if we take Bennu as an example asteroid that is still only 0.00006% of the total mass.
So we have moved thousands of tonnes of debris into the space around our target making it rather hazardous, but only moved a relatively small portion of the overall asteroid so not necessarily useful if trying to clear overburden off a metal core or trying to fully 'pit' the asteroid by ejecting the core entirely.
Separate to the questions around getting nuclear weapons for this purpose, a crater made this way will tend to expose frozen volitiles to sunlight which will probably tend to outgas long after the blast keeping the crater area populated with dust and debris, a more controlled mechanical excavation may reduce this, or even allow them to be captured.
There may be mining situations where this makes sense, but it is unclear at this time what that would be - if we are after HE3 from the solar wind then we want to carefully strip the surface off without disturbing anything.
While I'd have to check exactly what yield you would need you could probably expose the core of an asteroid fairly easily. However, you run into several massive issues.
the outer space treaty forbids the use of nukes in space
you would create a vast amount of space junk of a whole range of sizes. This space junk would spread out in all directions and make it increasingly risky to approach the asteroid. Additionally given the very large energies involved the space junk would fly out to a range of nearby orbits.
you've just made everything you want to mine fairly radioactive
Finally if you're wanting to mine the core of the asteroid then you're going to need mining equipment capable of cutting through whatever the core is made of. That means you already have the equipment to cut strait through the outer parts of the asteroid and reach the core, in which case why not simply do that?