During the Southern Song Dynasty, there was quite a bit of economic restructuring and growth. I'm aware that Song coins became more common in East Asia, but how much of an influence did they have in Japan specifically? I'm not familiar with Japanese currency, so sorry if this is an obvious or dumb part to the question, but to what extent did the coins see use or influence Japanese currency? Also, did any of the tested paper currency at the time make it over or have any influence (there's less records of the paper currency leaving China)?
The Southern Song Dynasty was the first to issue government credit notes called Hue-tsu notes. These held their value initially. During the later years of the Sung dynasty the quantity of hue-tsu issued was ever increased to the point where the country became inundated with paper notes. Over several decades the value of hue-tsu fell and at the end of the dynasty they had become almost worthless. Although, numerous other government issued credit notes appeared throughout the dynasty. Many of these were for military expenditures or for commodities such as salt, rice and tea. (Very Helpful Chinese Paper Money PDF)
Like those before them the Song Dynasty had used coins with square holes. However, there were vast amounts copper were mined and minted. So much copper was minted into coins that the price of copper rose drastically.
Japan did not issue paper money until 1800s, but there were also private notes issued in the 1600s, but I would not consider there to be a link between theirs and the Chinese. Chinese coins, on the other hand, were very much so used. Around the late 900s Japan strength weakened and they were no longer able to mint their own coins. Japanese officials then decided to import Chinese coins.
TL;DR Chinese paper money influence Japan, no way. Chinese coins influence Japan, absolutely.