The LDS Church does believe that Heavenly Father and Jesus Christ have the ability to cure and illness of injury if that is part of there plan for our eternal salvation.  
 
That does not mean though that every time someone gets sick or injured that they will miraculously heal such things would frustrate the plan of salvation as well as the purpose of moral life (for us to experience and grow).  
 
We believe that the understanding, training, and skill of physicians are also blessings from God and that since he has blessed these men and women with the ability to help aid the body's natural healing process that we cannot demand of God to heal us when he has already provided physicians to help us in these things.  
 
Members see the physicians of there choice as well as receive priesthood blessings of comfort and healing. We believe that the Lord is there to help us endure the trials of sickness and the limitations of mortality, we also believe that even when science says there is no hope that the the Lord can still bring full recovery if that is what is needed for his plan to go forward.  
 
In the end we believe the tGod can heal if that is what is needed, but we are to turn to the knowledge we already have first instead of expecting God to do everything for us!  
 
For official answers to questions like this I encourage you to visit LDS.org or mormon.org, understand that these responses are my personal interpretations and DO NOT necessarily represent the Official Position of the Church of Jesus Christ of Latter-day Saints!

One of the best summaries on this topic came from Brigham Young: 
 
"If we are sick, and ask the Lord to heal us, and to do all for us that is necessary to be done, according to my understanding of the Gospel of salvation, I might as well ask the Lord to cause my wheat and corn to grow, without my plowing the ground and casting in the seed. It appears consistent to me to apply every remedy that comes within the range of my knowledge, and to ask my Father in Heaven, in the name of Jesus Christ, to sanctify that application to the healing of my body. 
 
"But suppose we were traveling in the mountains, … and one or two were taken sick, without anything in the world in the shape of healing medicine within our reach, what should we do? According to my faith, ask the Lord Almighty to … heal the sick. This is our privilege, when so situated that we cannot get anything to help ourselves. Then the Lord and his servants can do all. But it is my duty to do, when I have it in my power. 
 
"We lay hands on the sick and wish them to be healed, and pray the Lord to heal them, but we cannot always say that he will." 
 
Source: www.lds.org/manual/teachings-brigham-young/chapter-34