Delete the first periodic digit
Wolfram Language (Mathematica), 59 bytes
FromDigits@MapAt[RotateLeft@*List@@#&amp;,RealDigits@#,{1,-1}]&amp;

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Explanation
RealDigits@#

Find the decimal digits of the input.
MapAt[RotateLeft@*List@@#&amp;, ..., {1,-1}]

If there are repeating digits, RotateLeft them. (List@@# prevents the code from attempting to rotate the last decimal digit if the rational number is terminating).
FromDigits@

Convert to rational.