Is there a setting or plugin to make WaniKani automatically play the pronunciation of a word after I get the Japanese reading right? I feel like this would help aid my listening and speaking.
Yep! Click your little picture on the top right → app (under settings) → autoplay audio in reviews. You can also set a preferred voice with the first option, too.
You may also be interested in the self-study quiz which can also be configured to play the audio for vocabulary words, both after providing a correct reading and before to actually quiz you on audible readings. The latter isn’t possible with Wanikani alone.
I highly recommend configuring a quiz for vocabulary in stages 1 and 2 using the Vocabulary audio → reading and Vocabulary audio → meaning settings. It may help significantly in your ability to hear and understand spoken Japanese if you practice newly-introduced words this way.
The human brain is freakishly good at recognizing patterns if you give it enough data (practice/repetitions). Limiting it to just early SRS stage data avoids most of the synonym/homonym problems.