City face Chelsea in Saturday’s League Cup final (12:15 GMT kick-off), which is live on BBC One, in a run of four back-to-back games against the Blues.
They will then play Sonia Bompastor’s side twice in their two-legged Champions League quarter-final, with a WSL meeting in between.
Taylor had been a controversial figure at City, with several players speaking out publicly about their experiences during his time in charge, including England goalkeeper Ellie Roebuck describing her relationship with him as “fractured”.
However, Greenwood said she had a “really positive” relationship with the 52-year-old.
“I was his captain and we had a really healthy relationship,” she added.
“I don’t expect everyone to have the same relationship with the manager, that’s football. Ultimately he has to make decisions and he’s never going to please everyone.”
Cushing will be in charge for the remainder of the season, having previously helped City win the WSL and the FA Cup, plus the League Cup twice, before leaving in January 2020 to join MLS side New York City.
“We were excited by that appointment because he knows the club really well, he’s been here before, he comes with a huge history of winning and his mentality to win is second to none,” said Greenwood.
“I had a really, really good chat with him on Tuesday morning.
“He had a great meeting with the team – a really, really positive one. It was ultimately about winning and what is achievable, what we need to focus on, which, first of all, is Saturday.”
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