Hah! I know this one because I've answered it before at Manga about a man who is repeatedly transported to a fantasy setting, where he helps the people he meets.
Part Time Overworlder by Amaou
Meet Tanaka. He’s a 26-year-old freeter - a guy who makes his living on various part-time jobs. He works in warehouses, directs traffic, and generally does whatever work he can find in order to get paid.
However, recently Tanaka has had a problem. He occasionally is transported into another world, usually in front of people who need help. Knowing nothing, can Tanaka add another part-time job to his repertoire?
The first time, he thinks it's just a dream, distracting the monster with some grass, bobbing it like a cat toy.
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It's an exhausted sorcerer (out of MP) leaning against a rock in the forest that he feeds the soup that he accidentally got from the vending machine instead of the beverage he was intended.
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It's then a cave lizard that he accidentally feeds work snacks to.
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Later, we meet his cat, and he meets the cat monster again, accidentally distracting it with the snack he was trying to feed his cat.
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And indeed, the last translated chapter I've found (chapter 18), it's implied that his cat, Eyebrows, is a "light god" who ran away to get a break from prayers, with the "dark god" (an identical cat but with dark fur) having also shown up.
Found by remembering that I might have answered this before, and searching for user:23243 [manga] jobs
, which looks for questions that I've asked or answered in the manga category and which have the word "jobs" in them. :-D