Numbers of right and left gloves Question? I assumed that numbers of right gloves at the horizontal axis and numbers of left gloves at the vertical axis. Therefore, two goods are essential and indifference curves are L-shaped. However, I want to ask that if my left hand is broken and I cannot wear the left glove for 12 months, does the indifference curve look like https://ibb.co/BfmF53H  Also, what can I say about its utility function? Can I say that its utility function looks like $U(x,y) = x^\alpha$
I don't think it is appropriate to apply the such static visualization to dynamic problem. Let us for a second forget about the graph - the graph is just a visualization of the indifference curve.
If two things are perfect complements which are consumed in 1:1 ratio their utility is given as:
$$u(x, y) = \min \{x, y\}$$
In this case (before you break the arm) the indifference curve will indeed look like L. If you break your arm the utility function would have to change but if we assume that the second good $y$ does not give you any utility it will just default to some utility of one variable $u(x)$.
However, for the $u(x)$ we cannot meaningfully plot an indifference curve and neither we can do it for $U(x,y) = x^{\alpha}$. Why? Because indifference curve is by definition a curve along which utility of consuming some combination of $x,y$ is constant. If $y$ does not gives us any utility then the person will always just consume maximum $x$ possible. Why should the person even care about $y$ at all? In such case the concept of indifference curve becomes meaningless as there is nothing to be indifferent about. You always just consume as much $x$ as possible.