I thinkboth Sleepy Kitty and Penguin make good points.
I think I would go with something like:
Thank you for your e-mail, in which you asked how you can increase the number of shifts you work, and why you did not receive some of the additional shifts you requested.
Please bear in mind that you are not being denied shift, other people are being approved. We have to try to select the most appropriate person for each shift, which includes looking at factors such as skill level as it relates to the shift, potential for overtime, and general availability.
During your meeting with [Bossman] and me when you were hired, you were told that given your limited availability, you were only likely to receive about 2 shifts a week.
You are correct that you were encouraged to pick up shifts, but as you recognise, lots of other employees are also applying to pick up those extra shifts. Having checked the records, it looks as though you only applied for 4 extra shifts during September, out of 60 shifts which were available. If you are able to apply for more of the available shifts, and to apply for shifts more evenly through the whole of the month instead of waiting until the last week of the month you will increase your chances of being approved for extra shifts. However, as he decisions about shifts are made based on what is best for the business, We cannot guarantee that you will be approved
Please feel free to contact me with any other concerns.
If there are specific areas where her skill are lacking, then I would add in something to make this clear, and to explain what she needs to do to change that - e.g. "At present,. you don't have experience of [relevant skill/skills] and you haven't yet reached the highest standard in relation to [other skills] If you wish to learn [relevant skill] you would need to speak to me or Bossman about [whatever training would be necessary]
I think it is legitimate to raise the issue about her e-mail but I would do that in a separate conversation, as it isn't directly relevant to the issue which she has raised (I assume that sending e-mails isn't part of her job?)