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The answer is me.
In the given sentence, "me" is in objective case of pronoun.

Putting this in sentence pattern/frame perspective:
Subject - Transitive Verb - Indirect Object - Direct Object (S-TV-IO-DO).

Cora - Subject
handed - transitive verb
Tom and me - indirect object  
        (the pronoun "me" must be used because it is an object pronoun,
       while "I" is a subject pronoun.)                                             
applications - direct object.