Andy Parsley's Answer:
To me this is a situation which requires similar thinking to the martial art of judo.
Your "senior opponents" in this situation are bigger and more powerful than you are, and if you try to take them on through direct confrontation you will not succeed. However, by channelling their strength and momentum in the direction you want to go you might just get to where you want to be.
So ask yourself (1) what do these senior people want to achieve for themselves? (money, power, security, status ... whatever), and (2) what are the issues they have to address to achieve these things? (in this case it's probably solving the "urgent operational matters" you have identified).
Think about these questions long and hard. Then identify those issues where you believe HR can make a real difference and the senior individuals who are likely to derive the greatest personal advantage by successfully resolving these issues.
You then need to demonstrate to your targeted senior executive(s) exactly how HR and Training interventions will solve their problems (ideally in a way that's better, simpler or cheaper than the alternatives) and, critically but discretely, how supporting the implementation of these initiatives will give them the personal outcomes they are after.
Naked self-interest will look after the rest - all you'll need to do then is deliver on your promises!
As for the "incompetent" group, with the above in place you'll simply need to demonstrate how they will potentially get in the way of solving the problems and they will soon get moved on. Being harsh, the most likely reason they were foisted on HR in the first place is that the function is currently perceived as being so ineffectual that it's somewhere where these misfits can be put where they won't do any damage. Prove HR's value to the organisation, however, and they'll be gone.
Identifying "areas for improvement" that senior managers place little importance on makes ANY function a nuisance in their eyes (and, worse than that, a nuisance that clearly doesn't understand the "real" issues the organisation is having to wrestle with).
On the other hand, a function which clearly understands the issues and which can help address these whist making the senior team look smart are an asset which will be invested in and nurtured. Which of these are you now? And which of these would you like to be?