Yes, the details of your self-employed income & expenses section should accurately report this situation.
To start, you must report the full amount of Form 1099-NEC. (For example 100 honorarium + 10 travel) The IRS received a copy of Form 1099-NEC and will look to match this on your return.
Next, report the travel expenses as usual in full. TurboTax will subtract it for you from income. Do not otherwise adjust the travel for being paid privately and reimbursed. (as an expense, this would be -10, which is the same as -10 +10 -10)
The result would appropriately not allow for an expense that was reimbursed, but not "overcharge" for the additional income.
For more information, see: Tax Topics for Freelancers, Contractors, and Consultants