Getting on with people long term is often about making them feeling acknowledged and being clear about what makes them valued.
The real trick to 'repair' is not to make hollow promises. Managers can be perceived as failing when an external event happens and they haven't planned for it, or they bet against it happening. This can kick off a whole chain of events, including pushing team members into crunch time or 'impossible positions'. Its rare that you can stop the external event or a similar one from happening, so promising it won't is hollow.
The next hollow promise commonly made is 'when it happens I won't let X happen [to you]'. The problem here is often that you probably will. In two ways: either X happening is clear in hindsight but not with foresight, so you'll probably make similar decisions again; or, the team member ending up in an unhappy situation is the best of a bad bunch of options.
I've had to place people in positions where they had insufficient support and excessive demands. Sometimes I knew this going in, and sometimes I did not.
You also have to be careful about passing the buck - if you're the manager you need to be clear with yourself about what your job is and whose issue any given problem actually is. Do you help your team interact with third parties, or do third parties interact with your team through you? How much are you supposed to represent your teams needs to management (e.g. pushback) vs how much are you supposed to represent your management's desires to the team (e.g. pushdown).
If you are caught passing the buck to shirk responsibility by your reports or by management you will lose a lot of trust and respect very quickly. You can always pushback or pushdown harder to appear 'good' to one party, but at some extreme that is going to lose you your job. Its your choice how to play this - so own the choice.