How can one purify speech? Speech or tongue is part of indriyas and mother Saraswati is the owner of it. So, how to purify one's speech?
One can either do Japa or praise Mother Saraswati or Lord Shiva to purify speech. Pushpadanta had purified his speech by singing Mahimna Strotra as he was Gandharva.
मधुस्फीता वाचः परमममृतं निर्मितवतस्तव ब्रह्मन्किं वागपि सुरगुरोर्विस्मयपदम्। मम त्वेनां वाणीं गुणकथनपुण्येन भवतः पुनामीत्यर्थेऽस्मिन् पुरमथन बुद्धिर्व्यवसिता॥३॥
Oh Brahma --the Absolute-- (brahman)!, is it (kim) surprising (vismayapadam) that even (api) the guru --preceptor-- (guroḥ) of the gods (sura) --i.e. "Bṛhaspati"-- (had uttered) word(s) (vāk) (in praise) of You (tava), who are the author --nirmitavān-- (nirmitavataḥ) of the Supreme (paramam) Nectar (amṛtam) --i.e. "the Veda-s"-- which consists of words (vācaḥ) abounding (sphītāḥ) in the honey (of knowledge) (madhu)? "I purify (punāmi) my (mama... enām) speech (vāṇīm) indeed (tu) by means of the meritorious act (puṇyena) of describing (kathana) Your (bhavataḥ) qualities or attributes (guṇa)"... for this purpose (ityarthe), (my) intellect (buddhiḥ) has resolutely undertaken (vyavasitā) this (task) (asmin), oh Destroyer (mathana) of the (three) cities (pura)! --i.e. Śiva is called "puramathana" because He destroyed the three cities belonging to the three sons of the demon known as Tāraka--||3||