Internal communications should be done by HR or by Corporate Communications team? Who can handle it better?

Both HR and the corporate communications team need internal communications but should be handled most effectively by the corporate communications team as experts in communication content developers but should be directed by the HR unit that oversees the entire people management of the organisation that drives communications needs.

It should be the job of HR in coordination with corporate communications team

I would say it depends on the communication! If it is something that directly impacts the employee, such as a policy relaunch or introduction of a new rewards scheme etc then probably best for HR to compose the communication, however, if it's in relation to the business performance generally then it would make more sense for the marketing team to lead on the communication. 

HRs after consulting Corp, communication team

Doesn't matter. 

What does matter is that the person in charge of any particular communication is able to explain in one sentence how this communication makes the company money.  If they can't, don't send it: they are just wasting the front line employees' time.