A smart city is an interconnected entity of internet of things and intelligent systems to provide qualify services to its citizens in various sectors such as public safety, healthcare, transportation and energy. In order for users to get the provided services, they have to access these services, provide and exchange a lot of personal information in an open, unsafe and interconnected environment. Therefore, user information may be exposed to privacy violations, which is especially critical for healthcare related services. Privacy engineering challenges that are not considered in design and development of these services may lead to possible privacy holes and leakage points, which raise concerns among users. In this paper, a privacy engineering approach and safeguard framework are proposed for smart city healthcare services. The paper aims to understand the privacy holes, identify the privacy leakage points and study privacy requirements to propose a privacy safeguard design solution.