Is it permissible to secretly watch anime without the mother's knowledge, while avoiding bad scenes, and is that considered a sin, and what is the Shar'i (Islamic legal) ruling on a mother preventing the watching of anime?

There is no harm in watching anime in principle, provided it is free of Sharia prohibitions. However, obeying one's mother is obligatory if she prevents you from watching it, especially since she has a valid reason, such as her fear that you might fall into the forbidden due to it containing bad scenes. Obedience is obligatory outwardly and inwardly, and this prohibition is a kindness from her to protect you from forbidden things and wasting time. Obey your mother's command out of obedience to Allah and out of filial piety towards her.