Expeditiously means quickly; expediently implies a quick fix that is in the immediate best interests of the provider but not necessarily those of the client.
Merriam Webster defines expedient as:
1 : suitable for achieving a particular end in a given circumstance
2 : characterized by concern with what is opportune; especially : governed by self-interest
Expedient, in current use, almost always carries that derogatory connotation. So if a service provider told me he’d deal with all issues expediently, I’d be inclined to take my business elsewhere.
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