If you're looking for manga similar to Inugamihime no Shimobe, you might like these titles. All recommendations are made by Anime-Planet users like you!
Nanami used to live in a ramshackle apartment with her father - a man who spent half his time gambling and the other half hiding from debt collectors - until one day the scoundrel ran away, leaving his daughter hungry and homeless. That is, until she meets a mysterious stranger in the park who deems her the new goddess of a local shrine! Unfortunately, Nanami’s new home is occupied by Tomoe, a fox familiar who objects to the presence of a mere mortal and immediately begins to bicker with the girl. If she can manage to form a contract with the troublesome man, he’ll be obligated to accept and help her perform her duties. But there’s a catch: Nanami has to kiss him first, and who’d want to lock lips with that awful guy?!
Both series take place in modern Japan and both of them are about a goddess and her servant. The thing is in Kamisama Hajimemashita the godess is actualy human and her servant is a youkai, but in Inugamihime no Shimobe the servant is human and the goddess is a youkai. If you want to read a light, japanese-lore-themed romcom, then both of those series should suit your taste.
Insecure, socially-awkward and with a habit of making sudden and profuse apologies, Asumi Torii is a strange girl who often fails to click with her peers... she also has an arrow stuck in her head. For new transfer student Ryouji Yamada, however, Asumi is the most interesting girl he's met. Simultaneously fascinating and unsettling though she may be, he dedicates himself to getting to know her and finding out what makes her tick. However her bizarre outlook on life - combined with the razor sharp arrow protruding from her head - makes it a painful learning experience for Ryouji.