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Civil court cases (particularly a breach of contract case) are not going to be removed. Matters that occur in our judicial system are public record. (we do this to prevent the exercise of our judicial system in private or secrecy, and to ensure that our system is fair to all). There are very few instances where a party can request that certain documents be redacted or quashed from the public record (for example cases where a witness (not a party to the case) has claims of abuse or some similar issue and they want those records pulled). In such cases they can make a motion to the court asking that the court seal or redact the records, but this is "extraordinary" (meaning it happens only in unusual circumstances for very compelling reasons).
A breach of contract claim is not going to rise to this standard.