Fuel injectors make a ticking noise, Fords(mazdas) are quite loud.
'99 4.0l would need 60psi fuel pressure at the injectors, less would cause computer to open them longer and could make them louder.
Also lower or varying pressure would make engine/computer struggle to try and keep air/fuel mix correct, so bucking, like you would get with dirty MAF sensor.
When you change the fuel pump you will get air in the system, no way around it, but it will come out at the injectors within a few minutes, this would cause stumbling engine and maybe injector noise.
But it would be purged after that.
If you didn't get fuel line connections tight or new fuel pump is not working right(very common now a days) then issues could continue.
The fact that adding more fuel changed engine operation would point me at new fuel pump issue.