I've heard sound like: Haru na kimasu. So why is that?
I'm going to be very frank here. I think it's because you're not yet able to distinguish intervocalic [ŋ] from [n] and [m]. Incidentally, listening to the passage you link to, I hear [ŋ] in the first, [ɡ] in the second ocurrence of しごと.
Do the Japanese not like the g sound (g as in gorilla)?
It's not that they don't like it. But the fact is that the phone [ɡ] is just one of the ways that the phoneme /ɡ/ is realized in Japanese. Have a look here. Often intervocalic /ɡ/ is realized as [ɣ] or [ŋ]. Also, intervocalic /b/ is often realized as [β], but this might not catch your attention as much, since this is less likely to cause confusion with other consonants.
If by the first question, you meant to ask why /ɡ/ it is pronounced differently from [ɡ], one way to answer that would be that [ɣ] or [ŋ] require less effort to pronounce intervocalically. [ɡ] is a stop, meaning that you have to stop the airstream while articulating, while [ɣ] or [ŋ] are a fricative and a nasal stop. In the former, the airstream is not stopped although the channel that the air goes through is made narrower. In the latter, the air stops going through the mouth and is directed through the nose.
This phenomenon, known as weakening, happens in other languages as well, e.g. in Castillian Spanish, where intervocalically (and in certain other positions) /ɡ/ -> [ɣ], /b/ -> /β/ etc.