Can I influence Places API results to a specific country?

Yes. Places API requests have component=country (Place Autocomplete) or region (Place Search, Place Details) parameters to influence the results to a specific country. These parameters take in a ccTLD (country code top-level domain) code which follows the ISO 3166-1 Alpha-2 standard so in the requests, countries need to be specified in its corresponding two-character code value. For example (value can be specified in all caps):

• us: United States
• gb: United Kingdom of Great Britain
• jp: Japan
• pr: Puerto Rico
• za: South Africa

Usage in Places API requests: 

Place Autocomplete component=country parameter usage: 

https://maps.googleapis.com/maps/api/place/autocomplete/json?input=Versailles&components=country:fr&key=<YOUR_API_KEY>

Place Details region parameter usage:

https://maps.googleapis.com/maps/api/place/details/json?place_id=ChIJDbg7ZMt95kcR_38nIKnyHmM&fields=name&region=fr&key=<YOUR_API_KEY>

Important: Before using any country code, please check if Google Maps Platform has coverage to its corresponding country. You can refer to Where can I check if a country has Google Maps Platform coverage? to get this information.