What do you call a switch/relay which switches on an appliance for a given set of time?
Those switches are called delay timer relays. 

For instance, see the Omron H3Y, sold on eBay for $8 or less:



Your application would require a relay with 10 minutes or more on the setting range, and you would need to ensure that the specific relay / relay base you buy are rated for inductive loads sufficient to cover the pump's starting / stall current rating.

Note that the sample image above is rated for 5 Amperes 250 Volts AC resistive load, not the rating applicable to your requirement.