This is strictly a flux and not a solder paste.
These are gums and resins- they are purified and mildly activated rosin.
These are soluble in organic solvent but insoluble in water. You can thin the paste with a little alcohol.
If the surface to be soldered is slightly dirty, solder will not stick. Common solder is an eutectic of lead and tin and they form an alloy with copper. That makes a strong bond.
Copper surface oxidises easily when exposed to air and that makes soldering difficult. The flux is a reducing agent and cleans the surface and helps the liquid solder to wet the surface.
Excess flux does no harm. You can also remove the excess left over on the board with ethanol or isopropyl alcohol.
Personally I feel that the solder present within the core is too little. You should use the paste every time you feel that the solder is not sticking well (within 2-3 secs after melting).
Apply solder flux directly on the board and then use the iron and the solder wire.
Usually it works well (unless the corrosion is extensive: it is clearly visible).