No. She can't carry forward unused tax relief from a previous tax year. What can be carried forward is any unused annual allowance from a previous pension input period. If the amount of contributions paid is more than the annual allowance available in the pension input period, she can carry forward any unused annual allowance from previous pension input periods to increase that year's annual allowance. Katie can go back as far as pension input periods ending three tax years ago.
See our article on Carry forward for further details.
If Katie has triggered the money purchase annual allowance, she can only use carry forward to allow her to pay more into a defined benefit plan.
If Katie is subject to the tapered annual allowance, it's still possible to carry forward unused annual allowance from previous years to a year where the taper applies. However, the amount of unused annual allowance available when carrying forward from a year where the taper has applied will be the balance of the tapered amount.
The annual allowance acts like a cap on the amount of tax relief an individual can receive. The tax relief is capped by imposing an annual allowance charge on any contribution over the annual allowance available for the pension input period. The charge is calculated using the individual's marginal rate of income tax, so it's designed to effectively remove the tax relief received on the contribution. This is done via Katie's tax return.
See our article on tax relief and the annual allowance for further details.