One possible plot could be that Zetsu, with his unique abilities, somehow crosses paths with Sakura. Maybe he's on a mission related to the Akatsuki and she gets in his way. He could initially view her as just another obstacle, but as time goes on, he realizes her strength and tenacity. This could lead to an unlikely alliance where they work together to achieve a common goal, like saving a village from a greater threat.
Zetsu could be sarcastic towards Sakura, taunting her about her human emotions. Sakura, in turn, would stand up for herself and show her strength. For example, Zetsu might say something like 'Your soft heart will be your downfall, little girl' and Sakura would reply 'My heart is what makes me strong, something you could never understand.'
Black Zetsu's story in Naruto is quite the tale. He was an entity that had been around since ancient times. He had an in - depth knowledge of the ninja world's history and used this to his advantage. He was the driving force behind many of the conflicts in the Naruto series. For example, he caused a rift between Sasuke and Naruto at times. His true nature was only revealed towards the end of the series, which was a shock to many characters. He was a master of manipulation, using the desires and weaknesses of others to achieve his ultimate aim of bringing back Kaguya.
One key event was his deception of Madara. He made Madara think he was his ally while actually setting him up for Kaguya's revival. Another was his influence on Obito. He pushed Obito to do things that served his purpose. Also, his role in the Fourth Great Ninja War was significant as he was behind many of the unexpected turns in the war.
It's likely that Naruto's chakra control and power would develop in new and unexpected ways. Black Zetsu might introduce him to strange chakra - based abilities that are not part of the normal ninja training. This could give Naruto an edge in combat, but it also might make him a target. Since Black Zetsu has his own motives, Naruto could be dragged into some dangerous and complicated situations that he may not be fully prepared for.
Perhaps a new, powerful and mysterious ninja who has a vendetta against the Hidden Leaf Village where Sakura is from. He sees Zetsu as an accomplice of the village and targets both of them, causing chaos and danger in their lives. This new character could bring in a fresh and unexpected element of danger to the fanfic.
Zetsu's ability to split into two parts can be used to create interesting plot devices. For example, one part of Zetsu could be captured by the protagonists, while the other part goes on to execute a different part of the villain's plan.
One idea could be that Zetsu discovers Sakura's hidden power during a mission, and they form an unlikely alliance. This alliance then leads to them exploring new areas together and facing various challenges as a team.