Is there a term for the indirect sponsorship of large corporates by the state? For example, unemployment welfare payments could be argued to be ameliorating the negative externalities of large corporates hiring newly arrived unskilled migrants.

Is there a term that covers this?
In European Union's nomenclature this is called 'State aid' (see glossary on EU's legislation site for a quick summary).
The idea behind this term is to cover all direct and indirect aid, including economic policy. If the measure is selective and it distorts the competition, it constitutes 'incompatible State aid'.

Besides the EU, I think the word you are looking for is 'subsidy', which covers both direct and indirect aid, or more generally, 'aid measure'.

Hope this helps, if still relevant to you :)