Is it permissible to work in an internet café that contains abominations such as songs, chat, obscene websites, and foreign films, given that the father prohibits leaving the job and threatens with his displeasure, and will his prayer against one be answered, and is the income earned permissible or impermissible?

You are not permitted to work in this cafe due to the assistance it provides in sin, and it is not permissible to obey your father by continuing to work there, because there is no obedience to a creature in disobedience to the Creator. A father's supplication against you for obeying Allah will not be answered because it is unjust.