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Yes, Mormons believe in gifts and miracles of both spiritual and physical healing in these days.
Elder Dallin H. Oaks of the Quorum of the Twelve Apostles gave a talk in 2010 called Healing the Sick. The whole talk is relevant to your question but here's a snippet (not sorry for making it essentially my whole answer):
Latter-day Saints believe in applying the best available scientific knowledge and techniques. We use nutrition, exercise, and other practices to preserve health, and we enlist the help of healing practitioners, such as physicians and surgeons, to restore health.
The use of medical science is not at odds with our prayers of faith and our reliance on priesthood blessings. When a person requested a priesthood blessing, Brigham Young would ask, “Have you used any remedies?” To those who said no because “we wish the Elders to lay hands upon us, and we have faith that we shall be healed,” President Young replied: “That is very inconsistent according to my faith. 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 [then] to ask my Father in Heaven … to sanctify that application to the healing of my body.”
Of course we don’t wait until all other methods are exhausted before we pray in faith or give priesthood blessings for healing. In emergencies, prayers and blessings come first. Most often we pursue all efforts simultaneously. This follows the scriptural teachings that we should “pray always” (D&C 90:24) and that all things should be done in wisdom and order.
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Truly, as the Book of Mormon teaches, God “manifesteth himself unto all those who believe in him, by the power of the Holy Ghost; yea, unto every nation, kindred, tongue, and people, working mighty miracles … among the children of men according to their faith” (2 Nephi 26:13).
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Many scriptures teach that the servants of the Lord “shall lay hands on the sick, and they shall recover” (Mark 16:18). Miracles happen when the authority of the priesthood is used to bless the sick. I have experienced these miracles. As a boy and as a man I have seen healings as miraculous as any recorded in the scriptures, and so have many of you.
You might say that the LDS believe in modern medicine as a miracle in and of itself. Elder Russell M. Nelson, another member of the Twelve, is a world-renowned heart surgeon who developed several new techniques for saving lives. But, as he has stated in his own talks on the matter, not without faith and prayers and fasting.
The Book of Mormon says, in Moroni 7:27,29,35-36:
27 Wherefore, my beloved brethren, have miracles ceased because Christ hath ascended into heaven, and hath sat down on the right hand of God, to claim of the Father his rights of mercy which he hath upon the children of men?
29 And because he hath done this, my beloved brethren, have miracles ceased? Behold I say unto you, Nay; neither have angels ceased to minister unto the children of men.
35 And now, my beloved brethren, if this be the case that these things are true which I have spoken unto you, and God will show unto you, with power and great glory at the last day, that they are true, and if they are true has the day of miracles ceased?
36 Or have angels ceased to appear unto the children of men? Or has he withheld the power of the Holy Ghost from them? Or will he, so long as time shall last, or the earth shall stand, or there shall be one man upon the face thereof to be saved?
37 Behold I say unto you, Nay; for it is by faith that miracles are wrought; and it is by faith that angels appear and minister unto men; wherefore, if these things have ceased wo be unto the children of men, for it is because of unbelief, and all is vain.
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Yes, mormons believe in physical healings. All Melchizedek priesthood holders–which constitutes most faithful men aged 18+ in the church–carry with them a small vial of oil, which has been consecrated for the healing of the sick and the afflicted. Blessings are given to those in need of physical healing, and there are no shortage of accounts or miracles of healing, although they are not generally publicized in the media. LDS consider such ordinances to be sacred blessings. I know several people personally who have been healed by priesthood blessings.
A young man I grew up with was hit by a car and pushed through a brick wall. Both of his femurs were shattered, and the doctors said he'd never walk again. He received a priesthood blessing that he would heal, and today he walks as though nothing had happened.
I know another girl who currently has doctors baffled. She was essentially crushed in a vehicle accident, broke many bones, including her skull. After narrowly saving her life, early scans revealed that she would at the very least be blind for the rest of her life, and that she would likely live out the rest of her life as a vegetable. She received a priesthood blessing that she would heal. After making a miraculous recovery, she went back to see a doctor whom she'd never seen before to assess her progress. When the doctor entered the waiting room to find her, she smiled and indicated she was the one the doctor was looking for. Noticing she wasn't in a wheel chair the doctor immediately asked her, "You walked in here?" After she replied that she had indeed walked in the doctor still didn't believe her, "Show me." he said. Having previously reviewed her charts and scans, the doctor initially refused to believe the girl he was looking at was the same girl whose brain scan he had previously viewed. Looking at her charts and scans he told her to her face, "According to what I'm seeing here, you can't walk, and you are blind." She is currently serving a proselyting mission for the LDS church.
Healings are common, but few are as miraculous as in Christ's time, though there are still some accounts. One of the more famous ones was by a man named John H. Groberg, who served a mission in Tonga in the 1950's. As a young missionary a boy was brought to him by frantic parents. This is from his own account of what happened:
One afternoon we heard cries of anguish and saw a family bringing the limp, seemingly lifeless body of their eight-year-old son to us. They wailed out that he had fallen from a mango tree and would not respond to anything. The faithful father and mother put him in my arms and said, “You have the Melchizedek Priesthood; bring him back to us whole and well.”
Elder Groberg participated in giving the boy a priesthood blessing, and the boy was restored to his parents.
Disney actually made a movie about Elder Groberg's mission called, "The Other Side of Heaven", but liberties were taken and they didn't portray the healing of the boy as Groberg recalls it in his memoirs. Guess it was too much for children's fantasy movie maker to portray a real miracle in one of their films.
Miraculous healing happens, and that is a fact accepted by all faithful members of the church.