Can You Skip Tarawih in Mosque If You Are Busy?
Question

I come back from job at 6.30pm & then take care of my child which is eight years old. My wife & younger son went back home to see her mother, who is in a life & death situation. In this case, I cannot go to the mosque for tarawih. Please guide me about this. I never go for tarawih but this Ramadan I had a plan. But unfortunately my wife went back for two months.
Praise be to Allah.Tarawih prayer is Sunnah for which the one who does it will be rewarded, but the one who does not do it will not be punished. So there is nothing wrong with your not praying Tarawih with the congregation . Allah knows the soundness of your intention and you may be rewarded even though you do not attend these nafl prayers, especially since you are not being negligent by not praying it.
Indeed sometimes a person may be kept busy with something that is more important, such as one who takes care of an elderly mother or orphan children who need to be looked after at such times. A person may be rewarded more if he is distracted from it by a good deed that needs to be done at that time. 
There is nothing wrong with praying Tarawih at home if you are unable to go to the mosque. 
It is not necessary to pray a specific number of rak`ahs , rather you should pray as many as you can manage. The Sunnah is to pray eleven rak`ahs.
If you pray two rak`ahs that is better than not praying. We ask Allah to compensate you with good for what you have missed and to record for you a reward that is greater than your praying Tarawih. 
Your letting your wife go and visit her mother and your taking care of your daughter are two great deeds for which you will be rewarded in sha Allah. 
But we would like to draw your attention to the fact that it is haram for your wife to travel on her own to your country without a mahram .
For more, please see this category: Tarawih Prayer and Laylat al-Qadr.
And Allah knows best.