A search for a pseudoscalar $a$ produced in association with a top-quark pair, or in association with a single top quark plus a $W$ boson, with the pseudoscalar decaying into b-quarks ($a\rightarrow b\bar{b}$), is performed using the full Run 2 data sample using a dileptonic decay mode signature. The search covers pseudoscalar boson masses between 12 and 100 GeV and involves both the kinematic regime where the decay products of the pseudoscalar are reconstructed as two standard $b$-tagged small-radius jets, or merged into a large-radius jet due to its Lorentz boost. No significant excess relative to expectations is observed. Assuming a branching ratio $\text {BR}(a\rightarrow b\bar{b})=100\% $, the range of pseudoscalar masses between 50 and 80 GeV is excluded at 95% confidence level for a coupling of the pseudoscalar to the top quark of 0.5, while a coupling of 1.0 is excluded at 95% confidence level for the masses considered, with the coupling defined as the strength modifier of the Standard Model Yukawa coupling.