Being a potential process that could explain gas phase abundances of so-called complex organic molecules (COMs) in the cold interstellar medium (ISM), the UV photon-induced desorption from organics-containing molecular ices has been experimentally studied. In this work, we focused on the observation of the photodesorbed products and the measurement of the associated photodesorption yields from pure and mixed molecular ices, each containing organic molecules whose detection has been achieved in the gas phase of the cold ISM, namely formic acid HCOOH and methyl formate HCOOCH3. Each molecule, in pure ice or in ice mixed with CO or water, was irradiated at 15 K with monochromatic vacuum UV photons in the 7-14 eV range using synchrotron radiation from the SOLEIL synchrotron facility, DESIRS beamline. Photodesorption yields of the intact molecules and of the photoproducts were derived as a function of the incident photon energy. Experiments have revealed that the desorbing species match the photodissociation pattern of each isolated molecule, with little influence of the kind of ice (pure or mixed in CO or H2O-rich environment). For both species, the photodesorption of the intact organics is found to be negligible in our experimental conditions, resulting in yields typically below 10-5 ejected molecules per incident photon. The results obtained on HCOOH and HCOOCH3-containing ices are similar to what has already been found for methanol-containing ices, but contrast with the case of another complex molecule, CH3CN, photodesorption of which has been recently studied. Such experimental results may be linked to the observation of COMs in protoplanetary disks, in which CH3CN is commonly observed whereas HCOOH or methanol are detected only in some sources, HCOOCH3 not being detected at all.