Oh, that's a good question! You're learning Ukrainian as well, aren't you? Then I'm sure you've noticed that this "softer х" sound exists in Ukrainian, too (and in Belarusian; sometimes it's pronounced ɣ, sometimes ɦ). It's also one of the features of Southern Russian used in the southwest of Russia, in Ukraine and Belarus*. In fact, it's used in Standard Russian as well but much less often: for example, I can pronounce the exclamation "о, господи!" ("oh God!") in two ways — with g or with ɣ — both are correct and common in Standard Russian.
As for the Moscow accent, hopefully drasvi will answer your question too, but the only thing that comes to mind is that in Standard Russian the unstressed vowel о is normally pronounced /ɐ/, while in the Moscow accent it may sound a little more like /а/ when this unstressed o is a pretonic vowel (for example, "московский" is pronounced /mɐˈskofskʲɪɪ̯/ in Standard Russian and /mаˈskofskʲɪɪ̯/ with a strong Moscow accent). This way of speaking is called сильное аканье (_а_канье). The difference between the Moscow accent and most other urban accents is quite subtle, though, at least nowadays; some people even say it's non-existent. Which is partly true, I think: most people can't tell whether a person my age or a little older is from Moscow or any other region where people speak Standard Russian based on their accent only (at least when there are no dialectisms used). Some people claim they can, but I can assure you that no one has ever managed to tell where I'm from based on my pronunciation (and there were many people who tried to make a guess), so I usually don't believe them. Maybe there are some phoneticians who can hear the difference, but in general, when it comes to standard urban Russian, regional differences get lost in individual differences (hope it makes sense).
/I'll get back to your question a bit later if you don't mind, there are some other things and examples I'd like to mention but I'll need my laptop for that//