If I have derivative asylum and have a green card, can I travel to home country for a month?

I would strongly advise against that as you can easily lose your asylum-based green card if you do so and it is discovered.  If you really want to see friends or relatives from your home country, see if you can meet them in a third  country so that you do not have to travel to your home country and risk losing your status in the US later on.  If you get an asylum in the US, that means you have shown that you are afraid of returning to your country because of fear of persecution on the basis of one or more established factors.  If you then voluntarily go to that country, that is an indication that you are really not afraid at all to travel there.