My question is that if one faces constant hardships, can it be a sign that Allah is trying to make him close to Him? Also, please advise any dua for one who is facing health issues one after the other?

Yes many times Allah is elevating His servant through hardships. You may see this lecture:

https://youtu.be/E9moJUeMJXg

Read this Du’a daily:

اللَّهُمَّ رَبَّ السَّمَاوَاتِ السَّبْعِ وَ مَا بَيْنَهُنَّ وَ رَبَّ الْعَرْشِ الْعَظِيمِ ، وَ رَبَّ جَبْرَئِيلَ وَ مِيكَائِيلَ وَ إِسْرَافِيلَ ، وَ رَبَّ الْقُرْآنِ الْعَظِيمِ ، وَ رَبَّ مُحَمَّدٍ خَاتَمِ النَّبِيِّينَ ، إِنِّي أَسْأَلُكَ بِالَّذِي تَقُومُ بِهِ السَّمَاءَ ، وَ بِهِ تَقُومُ الْأَرْضَ ، وَ بِهِ تُفَرِّقُ بَيْنَ الْجَمْعِ ، وَ بِهِ تَجْمَعُ بَيْنَ الْمُتَفَرِّقِ ، وَ بِهِ تَرْزُقُ الْأَحْيَاءَ ، وَ بِهِ أَحْصَيْتَ عَدَدَ الرِّمَالِ وَ وَزْنَ الْجِبَالِ وَ كَيْلَ الْبُحُورِ
Then say Salawat and ask Allah to help you with these health situations.

Also read this Du’a daily:

http://duas.org/Amn.htm

And this Dua:

http://www.duas.org/aafiyat.htm