The causal cognitive interference channel (CCIC) is a four-node channel, in which the second sender obtains information from the first sender causally and assists in the transmission of both. We propose a new coding scheme called Han-Kobayashi partial decode-forward binning (HK-PDF-binning), which combines the ideas of Han-Kobayashi coding, partial decode-forward relaying, conditional Gelfand-Pinsker binning and relaxed joint decoding. The second sender decodes a part of the message from the first sender, then uses Gelfand-Pinsker binning to bin against the decoded codeword. When applied to the Gaussian channel, this HK-PDF-binning essentializes to a correlation between the transmit signal and the state, which encompasses the traditional dirty-paper-coding binning as a special case when this correlation factor is zero. The proposed scheme encompasses the Han-Kobayashi rate region and achieves both partial decode-forward relaying rate for the first user and interference-free rate for the second user.