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The alternator wouldn't cause symptoms like you are having so nothing to worry about there.
The battery also shouldn't be an issue since it can't go completely dead and then back to normal like it would have to do to cause these symptoms.
The symptoms do point to a battery terminal connection or cable issue, and the fact that it almost invariably goes away when tapping on the battery terminals can't be overlooked. When trying to spread the original small terminals to match normal sized battery posts it often does cause connection issues and problems exactly like what you are having.
If you have enough cable you may be able to install common universal bolt on battery terminals. Otherwise replacing the cables/terminals along with the correct battery would be the first thing I recommend.
Thx for the reply. The battery is the correct battery. The problem is that the battery posts are just slightly too small, so the OEM terminals weren't good and snug on the posts. That is why I had to put on the lead 'cups' over the posts to make the structure big enough for the terminals to tighten up against for a solid connection. I never had this problem until I replaced the battery. I have been suspect of the lead cups, but they are both very tight now.
Please confirm you still would start with the battery terminals and cables.
Is this common in Honda Fits? I see comments that are all over the place online, but can't seem to find a commonly agreed upon answer. That is why I elected to pay an expert.