If the battery and connections are good and the starter is also good, more than likely the starter solenoid/relay is bad. A click usually indicates a loose battery cable or a faulty solenoid/relay.
When you hear a clicking noise is most likely a battery problem however you stated that the battery was replaced so i suggest you check the condition of your cables look for rust or corrosion etc.
battery need to be at lease 14 volts to be able to start
If there is clicking, that means the car has power from the battery so the altenator and the battery are good. I would suggest that the starter is gone bad. It will eventually grind away to where there is no clicking.
Could be the battery is drained or bad. Could be battery connections at battery and or starter are dirty and or loose. Could be starter solenoid is bad. Could be starter is bad.
either clean and tighten cable connections or get battery tested. if not the battery, it is the solenoid or the starter If it is clicking and not starting at first, but starts after a few tries, its your starter taking a dump. Replace it.
If it is a single click, the problem is either a loose battery cable or a faulty starter relay/solenoid. If the symptom is rapid clicking, the battery is probaly low.
It could be a starter problem....can you tell where the click is coming from? is it the started clicking, or a relay clicking?
dead battery bad starter solenoid bad starter loose or corroded battery or starter cables
Discharged battery.
Look for a loose or corroded battery cable. If the cables are ok at the battery, check the connection at the starter and at the starter solenoid. Otherwise, the solenoid may need to be replaced.
Bad Battery: the clicking is the starter solenoid relay making contact but the battery is not putting out enough amps to turn the starter
A single click, especially if it sounds quite heavy, is perfectly normal. This is just the starter solenoid engaging, switching the battery current to the Starter Motor. However a repeated clicking, usually indicates that a relay may not be latching correctly, consequently needs to be replaced. Check your Neutral Position Sensor - If the Vehicle is not fully in Park or Neutral it will not even attempt to start.