“We'll need to talk about this again. Can you manage a meeting next week?”
Does “manage a meeting” mean "manage to run a meeting"?
In this situation "manage" means "meet me for".
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Can you manage a meeting next week? = Will you be able to find the time to have a meeting next week?
AHD "manage", v.tr. def. 2. To succeed in accomplishing, achieving, or producing, especially with difficulty: managed to get a promotion; managed a polite goodbye.