To see Promotions management jobs, user should have access to LCM Folder. Check rights on this folder for your desired users group.
You mentioned that users are having issues running promotion management Jobs. Can you provide more details about the error message you are getting.
Are these users accounts mapped to AD or any 3-party authentication or Enterprise accounts?
Are they getting partial successful error when they perform promotions?
There are couple of KB articles on SAP support site to resolve partial successful error when 3rd party user accounts are used.