"Do you have a handkerchief?"
持つ【もつ】 is a verb meaning to hold, carry, possess - that sort of thing. もって is its て-form, used to combine it with other actions to form a sequence of things being done. In this case, we've combined it with います, the polite form of いる which if we're awkwardly literal gives "to hold and then be" but this is an idiomatic way of expressing a continuous tense - in English we might say "to be holding".
はしる - (I?) (will?) run.
はしります - polite form of above.
はしっている - (I?) am running.
はしっておよぐ - (I?) (will?) run and then swim.
はしっておよぎます - polite form.
Only the last verb needs to be made polite when you express a sequence of actions like this.
Not to lend to others, I hope.
Mind you, that wasn't necessarily the implication.