Would a direct hit by a short-period comet with a 3 mile diameter nucleus cause enough damage on Earth to threaten or destroy human life and the ability to produce crops?
Here, have some fun with this...
It doesn't go into long term effects, but the short term is pretty devastating for a significant portion of the planet. 'Nuclear winter' effects seem inevitable for those not atomised, scorched, crushed, or covered in ejecta.
Not good, nope.