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Rotating nonaxisymmetric neutron stars (NSs) are promising sources for continuous gravitational waves (CWs). CWs may, if detected, inform us about the internal structure and equation of state of NSs. Here, we present a narrowband search for CWs from known pulsars, for which a matched-filter search can be applied. Narrowband searches are robust to mismatches between electromagnetic (EM) and gravitational emissions, in contrast to fully targeted searches where they are assumed to be phase-locked. In this work, we search for the CW counterparts emitted by 34 pulsars using data from the first and second parts of the fourth LIGO─Virgo─KAGRA observing run. We use the 5n-vector narrowband pipeline, which applies frequency-domain matched filtering. In previous searches, it covered a narrow range in the frequency—frequency time derivative (f—<inline-formula> <mml:math><mml:mover><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>̇</mml:mo></mml:mrow></mml:mover></mml:math> </inline-formula>) space. Here, we also explore a range in the second time derivative of the frequency <inline-formula> <mml:math><mml:mover><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mo>̈</mml:mo></mml:mrow></mml:mover></mml:math> </inline-formula> around the EM observations. Additionally, for the first time, we target sources in a binary system with this kind of search. We find no evidence for CWs and therefore set upper limits on the strain amplitude emitted by each pulsar. For 20 analyses, we report an upper limit below the theoretical spin-down limit. The tightest constraint is for pulsar PSR J0534+2200 (the Crab pulsar), for which our strain upper limit on the CW amplitude corresponds to ≤0.04% of the spin-down power being radiated in the CW channel.