Well it's going to be difficult as you left it charging for a month.
1. Typically Androids get better battery efficiency over a few weeks of charge and discharge cycles algorithms learn your usage and turn of things that aren't needed. I wouldn't know if there are problems with leaving the phone plugged in a month that will affect this.
2. While in general it is ok to leave a phone plugged in since there is a battery protection circuit that cuts off charge, the question now is, where does the electrical flow go? Is there heat generated? Even if it's not overly hot, a battery constantly exposed to heat will occur degradation of held charge faster. Playing games while plugged in may also do this. It's the heat damaging the battery, not the charging. You left it plugged a month, so potentially it may have exposed to elevated temperatures constantly for a month.
3. Then we can't tell if there is a charging port or battery defect since its been plugged in for a month.
I say just use it for a week or so, take note of battery drop rates and post a battery usage graph.