Semenyo left Bournemouth for City at the beginning of January.
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Discussing their winter market campaign, Frank said on Friday: "The club is working tirelessly to do their best to improve the team, especially Johan (Lange), Fabio (Paratici), and Vinai (Venkatesham).
"The Lewis family is extremely committed to this project; they want to do everything."
Then he made a direct admission: "I'll only break my rule once: there's no doubt that it was clear the club wanted to sign Semenyo."
Frank continued: "We can't be too obsessed with the short term if it limits the future. This is the quality of players we're looking for.
"If we don't get it right now, it will be a great summer, and I have no doubt we'll see significant improvements."