Thank you in advance for you kind guidance.
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You should remember this set phrase as 手を抜く, which means "to cut corners" or "to get lazy". 抜く is a simple transitive verb that means "to omit" here. Its variations can be explained by the basic grammar rules.
In conclusion, 手は抜けない in your sentence is interchangeable with 手が抜けない and 手を抜けない (but not 手の抜けない). There is no big difference in meaning.