Is there one word that suggests that something was 'done out of anger'? Usage similar to something done deliberately or intentionally, only including the motivation of anger.
in an unkind way in order to hurt or upset somebody synonym maliciously ‘I don't need you,’ she said spitefully.
Note: 'out of spite' : a feeling of anger towards another person that makes someone want to annoy, upset, or hurt them, especially in a small way: He's the sort of man who would let down the tyres on your car just out of/from spite.
Regarding the context you provided in your comment "when a parent allows their child to do something. It's either generously granted, as in their divine will, or they permit it with sarcasm as if saying, "Go ahead, but just see what happens" , you could use "exasperatedly" :
"That child will be the death of me!" Despina complained exasperatedly as she pushed back her hair from her eyes.