Is it permissible to watch Islamic animated songs (anime), which may include images of girls and compilations from several films, provided they are free of indecent scenes, even if the idea may not be entirely clear but is close to the lyrics of the song?

There is no harm in listening to anasheed (Islamic vocal music) that is free of musical instruments. The mere presence of accompanying cartoon films does not prevent watching them, as watching cartoon films is permissible as long as they are free of Sharia-prohibited elements.