The pump doesn’t have a current spec which we need. Do you have a web site for the pump? From that I may be able to find a current draw indication somewhere. A google search for “Mini Submersible Pump 3-5V” turns up this one
which tells me what I need I expect, the pump needs about 200ma to run. A 12V wall wart should easily run that and a MOSFET module should be able to replace the relay. You will need to add a buck regulator to reduce the 12V to 5V for the pump but that is easy. The MOSFET board also removes the need for a driver (they draw less than 8ma of current.) A Fritzing part is available here
you would then need to find such a module to but but I expect that should be fairly easy. For the buck regulator there are two choices, first a fixed voltage one like this
and an adjustable one such as this
the advantage of the adjustable one is that you can set the output voltage via a screw driver adjustable pot on board. That would allow you to change the volume of water your pump outputs (at least between 3 and 5V, higher than that may damage the pump!) For the parts of the replies you don’t understand feel free to ask for clarification and I will try and explain. I don’t mind answering questions when something is unclear, in my opinion the only stupid question is the one you don’t ask because you think it is stupid 
. This stuff is generally fairly complex and has limitations (such as the Arduino current draws) that are not necessarily obvious if you don’t know about them.
Peter