Try installing latest npm version this happens a lot with ionic.
npm install -g npm@latest
npm cache verify
npm i
If this doesn’t work I advise you to switch to yarn. It’s faster & easier to use. This is the link to download YARN.
After installing just run this command yarn global add cordova ionic
Ionic CLI uses npm by default. So go to your project & run $ yarn install.
If you want Ionic to use yarn by default run this: $ ionic config set -g yarn true