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Discovering radio pulsars, particularly millisecond pulsars (MSPs), is important for a range of astrophysical applications, such as testing theories of gravity or probing the magneto-ionic interstellar medium. We aim to discover pulsars that may have been missed in previous pulsar searches by leveraging known pulsar observables (primarily polarisation) in the sensitive, low-frequency radio images from the Low-Frequency Array (LOFAR) Two-metre Sky Survey (LoTSS), and have commenced the Targeted search, using LoTSS images, for polarised pulsars (TULIPP) survey. For this survey, we identified linearly and circularly polarised point sources with flux densities brighter than 2 mJy in LoTSS images at a centre frequency of 144 MHz with a 48 MHz bandwidth. Over 40 known pulsars, half of which are MSPs, were detected as polarised sources in the LoTSS images and excluded from the survey. We have obtained beam-formed LOFAR observations of 30 candidates, which were searched for pulsations using coherent de-dispersion. Here, we present the results of the first year of the TULIPP survey. We discovered two pulsars, PSRs J1049+5822 and J1602+3901, with rotational periods of P=0.73 s and 3.7 ms, respectively. We also detected a further five known pulsars (two slowly-rotating pulsars and three MSPs) for which accurate sky positions were not available to allow a unique cross-match with LoTSS sources. This targeted survey presents a relatively efficient method by which pulsars, particularly MSPs, may be discovered using the flexible observing modes of sensitive radio telescopes such as the Square Kilometre Array and its pathfinders/precursors, particularly since wide-area all-sky surveys using coherent de-dispersion are currently computationally infeasible.