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The original Harris hawks optimization (HHO) algorithm has the problems of unstable optimization effect and easy to fall into stagnation. However, most of the improved HHO algorithms can not effectively improve the ability of the algorithm to jump out of the local optimum. In this regard, an integrated improved HHO (IIHHO) algorithm is proposed. Firstly, the linear transformation escape energy used by the original HHO algorithm is relatively simple and lacks the escape law of the prey in the actual nature. Therefore, intermittent energy regulator is introduced to adjust the energy of Harris hawks, which is conducive to improving the local search ability of the algorithm while restoring the prey's rest mechanism; Secondly, to adjust the uncertainty of random vector, a more regular vector change mechanism is used instead, and the attenuation vector is obtained by modifying the composite function. Finally, the search scope of Levy flight is further clarified, which is conducive to the algorithm jumping out of the local optimum. Finally, in order to modify the calculation limitations caused by the fixed step size, Cardano formula function is introduced to adjust the step size setting and improve the accuracy of the algorithm. First, the performance of IIHHO algorithm is analyzed on the Computational Experimental Competition 2013 (CEC 2013) function test set and compared with seven improved evolutionary algorithms, and the convergence value of the iterative curve obtained is better than most of the improved algorithms, verifying the effectiveness of the proposed IIHHO algorithm. Second, the IIHHO is compared with another three state of the art (SOTA) algorithms with the Computational Experimental Competition 2022 (CEC 2022) function test set, the experiments show that the proposed IIHHO algorithm still has a strong ability to search for the optimal value. Third, IIHHO algorithm is applied in two different engineering experiments. The calculation results of minimum cost prove that IIHHO algorithm has certain advantages in dealing with the problem of search space. All these demonstrate that the proposed IIHHO is promising for numeric optimization and engineering applications. [ABSTRACT FROM AUTHOR]