In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight w(x; t,α):= e-x2|x-t|α(A + B (x-t)),x (-∞,∞), where α >-1,A ≥ 0,A + B ≥ 0 and t It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381-12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo-Miwa-Okamoto σ form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the σ-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case A = 0,B = 1, we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight wα(x,t) = xαe-x2-2tx,x +, which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].