Pinyin "h" sound, pronunciation
In standard Mandarin, the pinyin h is pronounced as [x] in IPA, like ch in Scottish English loch -- yes, you're right that there's a tiny g, because [x] is a velar consonant. However, in south China, many simply say [h], although some natural assimilation may happen. For instance, for those speakers, pinyin ha is [ha] while pinyin he becomes [xɤ] (if you get [ɤ] right, then [hɤ] simply doesn't work, you would do [xɤ] unconsciously). Therefore [h] and [x] are allophones in this sense, so don't worry about your pronunciation.