In a small open economy specifically (as opposed to a closed economy, which does not trade with other countries, so $NX$ is not considered) , the the real world interest rate is not effected by the economy increasing its internal interest rate. As consumers can simply turn to the world market and get funds at a lower interest rate if the internal interest rate increases greater than the world interest rate. However an increase in government expenditure will increase the trade deficit.
Note that Net Exports equals Net Capital Outflow. This can be visualized mathematically:
Where: $S=$ National Savings $= Y-C-G$.
This means that as Government expenditure $(G)$ increases National Savings $(S)$ will decrease. Thus resulting in a greater trade deficit.
Therefore, It is of my humble opinion, based on the question, it seems that an increase in government expenditure will not assist in lowering the trade deficit.
Hope this is helpful.