Godan verb with 'ku' ending, Intransitive verb
to cling to ... in tears; to beg ... in tears; to appeal; to throw oneself on someone's mercy
案の定、あの業者今日になって泣きついてきたよ。Just like I thought, the contractor came back today, begging for more time.
Also written 泣付く【なきつく】、泣きつく【なきつく】