Have you now spoken to him?
I really heard that from someone.
There really is nothing wrong with it grammatically. But the context makes a difference on how apt it is.
The now implicitly indicates that there was a then. In other words, this is a continuation of a previous conversation (in most cases) where you perhaps mentioned that you wanted to speak to "him" or tried to speak to him and failed. The now questions whether you have been successful in speaking to him at this point in time.
A better example could be in a speech:
... and a few hours later
A variant of this usage could be somebody else situated elsewhere noting:
It's true that on average, "Have you now [done X]?" will be preferred over "Have you [done X] yet?", so non-native speakers who don't have a firm grasp of the distinction would be safest sticking to the former.
It's also true that in many contexts there's no meaningful distinction - so again, you may as well use the more common form.
Of course, there's no requirement to use either "temporal qualifier" - at the bare semantic level, "Have you [done something]?" is all you need to say.
But if "now" or "yet" is included, it emphasises the "temporal context". In the case of "now", obviously the focus is on whether the present time includes the attribute X has been done. With "yet" (or the alternative phrasing "Did you [do X]"), the focus is more on whether or not you did X at some point in the past.
In some contexts therefore, OP's usage of "now" can be seen as "deferential" and/or "confirmatory" in that it's more about objectively assessing the current situation, rather than subjectively judging the other person on the basis of whether he's got around to doing something the speaker expected (or indeed, required) him to do.
In one particular context, the handy pattern is:
"Yet" is an adverb that means "so far" or "up until now."
I suppose what you heard might have been an alteration of:
Some people, non-native speakers, also tend to mistakenly substitute "already" in the first example:
which means a completely different thing.