The 19-year-old French striker joined the newly promoted Premier League side on loan from Italian club AS Roma on transfer deadline day.
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"No individual should ever experience discriminatory abuse, and the club will provide Robinio with our full support," Hull said in a statement.
"We are also working with Humberside Police and other relevant authorities to identify the accounts responsible for this disgusting abuse, so that appropriate action can be taken against those individuals."
Hull have enjoyed a strong start to life back in the top flight, winning their opening two matches, and host Aston Villa on Saturday.