You did not mention the operating voltage and current of the gate opener and whether it is an AC or DC actuator. If it clicks, but does not buzz while the button is pressed it is DC operated. If it buzzes for the duration of the button press it is AC operated.
Since you mention that you are given this assignment in a work situation, I assume that time is an issue and an off-the-shelf solution makes more sense than a do-it-yourself hobbiest approach. True?
I would recommend a one-shot TDR (Time Delay Relay) as the most straightforward solution. There are myriad timing functions available in off-the-shelf time delay relays but the one you require is the one-shot function or the "Delay-On-Break" function. One-shot simply means that once the relay is actuated by the button push it stays actuated by itself for a preset period (usually, but not always adjustable) after the button is either pushed or released.
This type of TDR requires a continuous power supply to power its holding function after the button is released. It will have two pins or terminals that are permanently powered and (usually) two pins that connect to the pushbutton. Additionally, there will typically be three (for single-pole) or six (for double-pole) terminals that connect to the relay contacts. You only require a single-pole relay but may find that double-pole one-shot TDRs are most readily available.
You must match the relay to the power available at the gate and the current drawn by the gate actuator. You will find many socket-mounted TDRs offered with quite elaborate function selection switches and timing range switches. These are about US$75 and up. You don't need any of that. The cheapest devices look like black blocks with spade contacts sticking up and either terminals to which you solder a timing set resistor or a small knob to set the time delay. These are in the $20 - $30 range.
I don't know your location or your supply situation, but one US supplier is
www.grainger.com. Search for "Time Delay Relay."
You may also find that your local locksmith has special purpose timers for the very purpose of door/gate time delay and can install one for you.
Let us know what you find about your power and load and what approach you prefer.
awright