The timing belt and water pump should be replaced together probably. The pump was a $100 part and I had to pull the motor to get the timing cover off. My water pump was under the timing cover and driven by the timing belt. I took auto mechanics in school and they said to replace those 2 parts together because they last about the same amount of time. But I'm no expert.
If its not weeping you could re use it, it is typical for cheep insurance to do both but its an easy swap later, over added expence now, .... um lets consider the size and orientations of these engines ... im thinking chevy and you have hyundai, if its more than an hour to take the water pump off replace it while you there, if like my jimmy it takes 15 minutes or less with practice, i reuse
I am with Bofobo: if it’s easy to get at, just replace the timing belt and tensioner. If it’s a PITA, change the water pump as well.
As an aside: when we changed the timing belt on our Volvo @ 240K (it had been done once before by a Volvo Dealer garage @ 120K), we found they had missed the timing by one cog. It was three cogs out on a Subaru! So much for pro shops....
Not sure on your car, but some have the water pump driven by the timing belt. A water pump failure can cause the belt to fail and if the engine has interference valves, you will end up having to rebuild the head also. Cheap insurance to replace the water pump while doing the belt.
I priced out a head gasket replacement on a ford ranger...... 1600 labour on 400 parts..... waiting for spring to arrive and ill do it myself. Can't justify filling up the shop with the ranger in the winter