Hello and thank you for the opportunity to assist you. My name is ***** ***** I will do my very best to answer your legal questions.
The word "inadequate" simply has the ordinary English definition. The word "guardianship" comes from N.Y. Fam. Ct. Act § 1012. According to that statute, a parent is neglectful if that parent fails to provide the child with proper guardianship "by unreasonably inflicting or allowing to be inflicted harm, or a substantial risk thereof, including the infliction of excessive corporal punishment; or by misusing a drug or drugs; or by misusing alcoholic beverages to the extent that he loses self-control of his actions; or by any other acts of a similarly serious nature requiring the aid of the court; provided, however, that where the respondent is voluntarily and regularly participating in a rehabilitative program, evidence that the respondent has repeatedly misused a drug or drugs or alcoholic beverages to the extent that he loses self-control of his actions shall not establish that the child is a neglected child in the absence of evidence establishing that the child's physical, mental or emotional condition has been impaired or is in imminent danger of becoming impaired as set forth in paragraph (i) of this subdivision;"
So inadequate guardianship means that the parent does anything quoted in italics above. You'll note that includes "the child's physical, mental or emotional condition has been impaired or is in imminent danger of becoming impaired"
Obviously, that's quite a broad definition, but it's possible that a child's mental or emotional condition could be impaired (or is in danger of being impaired) if parents act violent towards one another while the child is in the same room. Please understand that I'm not judging you at all, but merely trying to help you to understand the situation.
Does that answer your question? Please let me know if you need clarification, as I am happy to continue helping you until you are satisfied.