Dear {Name},
I must start with an apology. It turns out that unfortunately I cannot follow through with {providing discount or service}.
My promise was made in haste, out of a sincere wish to provide you with what you sought. I should have checked with {boss/department} first. As it turns out, {it's against policy/we don't have the resources/it's illegal/it would put the project over budget}.
Here's what I'm able to do: {alternative offer}.
Again, I'm terribly sorry for breaking my promise to you. I deeply value our business relationship and I respect you as a person. I hope my misstep doesn't reflect poorly on our company as a whole. The mistake was mine and mine alone.
Sincerely,
{Name}
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Index of letter of apology templates