Pics are not very clear -
Around the mouth-Looks like perioral dermatitis with hyperpigmentation.
And Eczema on the palm looks a little infected.
Next Steps
Will need a proper history of symptoms and duration and any other comorbidity if any?
I would suggest making an appointment to understand things better with clearer pics and other needed details.
Health Tips
For mouth -You should avoid lip-licking if you are used to it.
And try switching to gel toothpaste
Use moisturizer frequently .
For palms - You should use a moisturizer frequently, avoid wet work and use silicone gloves if needed.