Any chance to apply for visa for child less than 1 years old?

Yes, you should apply for a Q1 type for your child. The required documents are:
(1)Foster entrustment notarization issued by Chinese Embassies/Consulates or Foster Care Power of Attorney notarized and authenticated in Singapore or China.
(2)Original and photocopy of the consignor’s passport(s), the original and photocopy of certification (marriage certificate, birth certificate, certification of kinship issued by Public Security Bureau or notarized certification of kinship) showing the relationship between parents and children.
(3) A letter of consent on foster care issued by the trustee living in China who has agreed to provide foster care services and a photocopy of the ID of the trustee.
(4) A photocopy of the certificate indicating the permanent residence status abroad of the parent(s) when the child was born, provided that either or both parents of the child are Chinese citizens.

PS: A passport for your child, completed application form and a recent passport photo are required too.